Cell Quota Based Population Models and their Applicationskuang/paper/AaronPHD.pdf · 2014-11-18 ·...

Cell Quota Based Population Models and theirApplications

Aaron Packer

School of Mathematical & Statistical SciencesArizona State University

November 17, 2014

A. Packer Cell Quota Based Models + Applications Nov 17, 2014 1 / 72

Outline1 Introduction2 Neutral Lipid Synthesis in Green Microalgae

IntroductionModelResultsConclusion

3 Applications to Stoichiometric Producer-Grazer Models for AquacultureIntroductionModelEquilibriaLinear Functional ResponseHolling Type II Functional Response

4 Prostate Cancer and Androgen Deprivation TherapyMechanistic derivationUptakeSingle population modelResultsFuture Work

A. Packer Cell Quota Based Models + Applications : — 2 / 72

Introduction

1 Introduction

2 Neutral Lipid Synthesis in Green MicroalgaeIntroductionModelResultsConclusion




Introduction

Cell Quota Model

In 1968, phycologist M.R. Droop published his famous discovery on thefunctional relationship between growth rate and internal nutrient status ofalgae in chemostat culture.

µ (Q) = µm

(1− q

Q

)

Q: cell quota (nutrient/biomass)q: subsistence quotaµm: maximum specific growth rate


Introduction

Cell Quota Model, cont’d

µ (Q) = µm

(1− q

Q

)

Leadbeater, B., “The ‘Droop Equation’–Michael Droop and the legacy of the ‘Cell- Quota Model’of phytoplankton growth”, Protist 157, 3, 345 (2006).


Introduction

Summary

Problems

Why do certain oleaginous algae produce so many neutral lipids, and how cantheir cultivation for biofuels be improved?

What role does ammonia- (and to lesser extend nitrite-) induced toxicity play inthe dynamics of producer-grazer systems for aquaculture?

Can cell-quota based population models be applied to prostate cancer in amechanistic way?

Solutions

The nitrogen cell quota quantifies the metabolic shift to neutral lipids in greenmicroalgae and gives rise to a mechanistic modeling framework.

Nitrogen toxicity can be modeled by adding new feedback into producer-grazersystems with nutrient recycling.

A mechanistically derived model follows naturally via application of the “cellquota” concept to androgens.


Neutral Lipid Synthesis in Green Microalgae

1 Introduction





Neutral Lipid Synthesis in Green Microalgae Introduction







Motivation

TheoreticalWhy do certain species of green microalgae produce such large quantities ofneutral lipids, particularly triacylglycerols, during stressed conditions?

MathematicalIs a mechanistic model possible, and what insight can be gained?



Motivation


Compensate for lack of electron/carbon sink during uncoupling ofphotosynthesis from growth.Transient energy storage during stressful times




Motivation


Compensate for lack of electron/carbon sink during uncoupling ofphotosynthesis from growth.Transient energy storage during stressful times


Yes!Mechanistic modeling helps validate current theory explaining the NLphenomenon in oleaginous algae



Key Observations - Nitrogen & Light

Nitrogen Starvation

Increases neutral lipid synthesis.Decreases cellular growth and production of non–neutral lipid biomass.Decreases capacity of certain mechanisms that prevent and repair bothphotoinhibition and photooxidation.

Light Intensity

Increasing light intensity...Increases neutral lipid synthesis.Increases susceptibility to photoinhibition and photooxidation.



Key Observations - Nitrogen & Light

Nitrogen Starvation

Increases neutral lipid synthesis.Decreases cellular growth and production of non–neutral lipid biomass.Decreases capacity of certain mechanisms that prevent and repair bothphotoinhibition and photooxidation.

↘↗⇒ NLs are defense from the dangers of photosynthesis–growth uncoupling

Light Intensity

Increasing light intensity...Increases neutral lipid synthesis.Increases susceptibility to photoinhibition and photooxidation.



Case Study: Pseudochlorococcum sp. (Li et al. 2011)

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

g N

/L

days

100%25%0%

Figure 1: Extracellular N

0 2 4 6 8 10 12

1

2

3

4

5

6

7

8

g/L

days

100%25%0%

Figure 2: Biomass

100% culture: 0.24 g N L-1. 25% culture: 0.06 g N/L. 0% culture: 0.0 g N L-1.

Li, Y., D. Han, M. Sommerfeld, and Q. Hu. “Photosynthetic carbon partitioning and lipid productionin the oleaginous microalga Pseudochlorococcum sp. (Chlorophyceae) under nitrogen-limitedconditions.” Bioresource Technology 102, 1 (2011): 123–129.




0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

g N

/L

days

100%25%0%


0 2 4 6 8 10 120

10

20

30

40

50

%

days

100%25%0%

Figure 2: NL % of biomass






0 2 4 6 8 10 12

1

2

3

4

5

g/L

days

100%25%0%

Figure 1: Non-NL biomass

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

g/L

days

100%25%0%

Figure 2: NLs




Neutral Lipid Synthesis in Green Microalgae Model







Model State Variables

A(t) = algae biomass density, excluding neutral lipids (g d.w. · m−3),

L(t) = neutral lipid density (g NL · m−3),

Q(t) = N-quota of A(t) (g N · g−1 d.w.),

H(t) = chl a content of A (g chl · g−1 d.w.),

N(t) = extracellular nitrogen concentration (g N · m−3).

A and LBiomass is divided into two compartments: non-NL biomass A(t) and NLs L(t).Therefore, total biomass density is the sum of the two compartments, A(t) + L(t).

Q and H

The N-quota, Q(t), is the intracellular N per unit A(t). Q(t)A(t) = total intracellular N.Similarly, H(t) is the intracellular chl a per unit A(t). H(t)A(t) = total chl a density.



Model

dAdt

= µA(t),︸︷︷︸cellular growth

(1)

dLdt

= (p− cµ) A(t),︸︷︷︸NL synthesis

(2)

dQdt

= v︸︷︷︸N uptake

− µQ(t),︸︷︷︸growth dilution

(3)

dHdt

= µ

p/cθmv︸︷︷︸

N uptake devoted to chl a synthesis

− µH(t),︸︷︷︸growth dilution

(4)

[dNdt

= −vA.︸︷︷︸N uptake

](5)



Growth rate

µ = min{µm

(1− q

Q(t)

),

pc

}

q1 minimum/subsistence N quota g N g-1 d.w.c C subsistence quota g C g−1 d.w.µm maximum N-limited growth rate s−1

p dw-specific photosynthesis rate (g C g-1 dw s-1)

Growth is either N or light limited.

N limited growth follows the cell quota model, µm

(1− q

Q(t)

).

Light limited growth is pc .

c is how much carbon is required per unit increase in dry weight.p is the photosynthesis rate.



Photosynthesis rate

p = H(t)pm

(1− exp

(−aΦI

pm

))

a absorption efficiency normalized to chl a m2 g−1 chlΦ quantum efficiency g C (µmol photons)−1

pm light-saturated photosynthesis rate g C g−1 chl s−1

The photosynthesis rate is modeled using the general Poisson model.Light-limited rate is governed by aΦI.Light-saturated rate, pm, is a function.



Light-saturated photosynthesis rate, pm

pm = p0Q2

Q2 + q2= (AQ)2p0

(AQ)2 + q2 (A(t) + L(t))2

q minimum/subsistence N quota g N g−1 d.w.p0 maximum photosynthesis rate g C g−1 chl s−1

Q N content relative to A + L g N g−1 d.w.

Previous models have assumed pm = 0 for Q = q, which does not workfor the NL model here.pm > 0 for Q = q in this model indicates decoupled photosynthesis fromgrowth.pm decreases as Q decreases.



NL synthesis

dLdt

= (p− cµ) A(t)

c C subsistence quota g C g−1 d.w.p photosynthesis rate (g C g-1 dw s-1)µ cellular (non-NL) growth rate s−1

NL synthesis results from an excess of C-fixation relative to the Crequirements for growth. cµ is C required for growth.For Q(t) = q ( ˜Q(t) ≤ q), all increases in total biomass are due to de novoNL synthesis.



Self shading and reactor depth

I = I0

aH(t)A(t)z(1− exp (−aH(t)A(t)z))

I0 incident irradiance µmol photons m−2 s−1

z light path ma absorption efficiency normalized to chl a m2 g−1 chl

I is average irradiance in the reactor.Derived using Lambert-Beer law of light attenuation.Enables model to incorporate self shading and to make qualitativelyaccurate predictions of biomass and NL dependence on z.



N uptake and Chl a synthesis

v = qM − Q(t)qM − q

(vmN(t)

N(t) + vh

)dHdt

= cµpθmv− H(t)µ

q minimum/subsistence N quota g N g−1 d.w.qM maximum N quota g N g−1 d.w.c C subsistence quota g C g−1 d.w.vm maximum uptake rate of nitrogen g N g d.w.−1 s−1

vh half-saturation coefficient g N m−3

θm maximum chl:N g chl a g−1 N

Chl a synthesis is coupled to N uptake/assimilation.Proportion of N uptake devoted to chl a synthesis is (cµ/p)θm.cµ/p represents the utilization ratio of fixed carbon.


Neutral Lipid Synthesis in Green Microalgae Results







Results

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

g N

/L

d

100%25%0%


0 2 4 6 8 10 120

1

2

3

4

5

6

7

d

g/L

25%0%

Figure 4: Total biomass A + L

Model fitted to data from Li et al. (2011). (100% culture excluded fromFigure 4.)



Results

0 2 4 6 8 10 120

20

40

60

80

d

%

25%0%

Figure 3: NL % of biomass

0 2 4 6 8 10 120

1

2

3

4

d

g N

L/L

25%0%

Figure 4: Neutral lipids L

Model fitted to data from Li et al. (2011). (100% culture excluded.)



Results

0 2 4 6 8 10 120

0.02

0.04

0.06

0.08

g N

/g d

w

d

25% Q0% Q

Figure 3: Q and Q

0 2 4 6 8 10 120

0.005

0.01

0.015

0.02

0.025

g C

hl/g

dw

d

100%25%0%

Figure 4: H(t), the chl acontent of A

Model fitted to data from Li et al. (2011). (Chl a data not reported.)


Neutral Lipid Synthesis in Green Microalgae Conclusion






Neutral Lipid Synthesis in Green Microalgae Conclusion

Conclusion

NL synthesis in oleaginous microalgae

Decoupling of photosynthesis from growth may explain NL synthesis.Minimum cell quota of limiting nutrient may represent a threshold for NLsynthesis.

100% (high-N) culture

Was N not the limiting resource for the 100% culture?Most likely given that a 4-fold increase in N resulted in 36% biomassincrease.

Future workSplitting biomass into separate compartments for functional biomass andneutral lipids is a useful framework, and has since been adopted in latermodels.Modeling neutral lipid synthesis is an active research area which hassince used ideas from the model presented here.


Applications to Stoichiometric Producer-Grazer Models for Aquaculture

1 Introduction





Applications to Stoichiometric Producer-Grazer Models for Aquaculture Introduction







Aquaculture

Farming of aquatic organisms.

Fish (Salmon, Carp, Grouper, Tilapia)Crustaceans (Shrimp, crab, prawn)Molluscs (Oyster, mussel)Aquatic plants (algae, seaweed)

MethodsPondsTanksRacewaysCages



Aquaculture

Farming of aquatic organisms.

Fish (Salmon, Carp, Grouper, Tilapia)Crustaceans (Shrimp, crab, prawn)Molluscs (Oyster, mussel)Aquatic plants (algae, seaweed)

MethodsPondsTanksRacewaysCages

Perfect for applied mathematical ecology.



Nitrogen & toxicity

Nitrogen cycle

An important part of aquaculture systems, with much existing research andeven mathematical models in the literature.

Toxicity

Accumulation of waste in culture.High levels of inorganic N can be toxic (ammonia, nitrite).Even low levels of ammonia or nitrite can have inhibitory effect on somespecies.

Producer-grazer modeling

What implications does N-induced toxicity have for N recycling and dynamicalbehavior in producer-grazer systems?


Applications to Stoichiometric Producer-Grazer Models for Aquaculture Model







Model

State variables

y(t) = grazer biomass density (g C · L−1)

x(t) = producer (phytoplankton) biomass density (g C · L−1)

Q(t) = N:C of producer (g N · g−1 C)[N(t) = external N concentration (g N · L−1)

]∗N(t)*System is assumed to be closed under nitrogen, so N(t) can be decoupledfrom the system.



Toxicity Model

x′ = m(

1− qQ

)x︸︷︷︸

N-limited growth

− f (x)y︸︷︷︸grazing

y′ = r min{

1,Qθ

}f (x)y︸︷︷︸

growth

− dy︸︷︷︸natural death

− h (N) y︸︷︷︸intoxication

Q′ = v(N)︸︷︷︸uptake

− m (Q− q)︸︷︷︸growth dilution[

N′ = −v(N)x︸︷︷︸uptake

+(

Q− r min {θ,Q})

f (x)y︸︷︷︸grazer waste

+ dθy + h (N) θy︸︷︷︸grazer death

]



Toxicity Model

x′ = m(

1− qQ

)x︸︷︷︸

N-limited growth

− f (x)y︸︷︷︸grazing

y′ = r min{

1,Qθ

}f (x)y︸︷︷︸

growth

− dy︸︷︷︸natural death

− h (T − Qx− θy) y︸︷︷︸intoxication

Q′ = v(T − Qx− θy)︸︷︷︸uptake

− m (Q− q)︸︷︷︸growth dilution



Parameter Description UnitsT total system nitrogen g N L-1

m producer maximum growth rate d-1

q producer minimum N:C quota g N g-1 Cθ grazer homeostatic N:C g N g-1 Cr grazing/digestion efficiency scalard grazer natural death rate d-1

Function Description Unitsf (x) functional response g C g-1 C d-1

v(N) producer-specific N uptake rate g N g-1 C d-1

h(N) grazer toxicity death rate d-1

Model parameters and generalized functions.f (0) = 0, f ′ > 0, f ′′ ≤ 0, and similarly for v and h.


Applications to Stoichiometric Producer-Grazer Models for Aquaculture Equilibria







Equilibria

Boundary equilibria

There are two unique boundary equilibria:

E0 = (0, 0, q + v(T)/m) (Extinction)E1 = (T/q, 0, q) (Grazer-only extinction)

Interal equilibria

Depending on f and the parameter values, there may be zero, one unique, ormultiple internal equilibria. It is not possible to find explicit formulas for theinternal equilibria in general.

E2 = (x∗, y∗,Q∗) (Coexistence)



Boundary Stability

Total extinctionE0 is always unstable.

Grazer-only extinction

E1 is locally asymptotically stable if and only if

rf (T/q) < dθq.

E1 is globally asymptotically stable if

rf (T/q)d

< 1 orrTf ′(0)

dθ< 1.



Internal equilibira

The coexistence equilibria E2 = (x, y,Q) are given by values Q which satisfy

T = N(Q) +[

Q + rAm(Q− q)d + h (N(Q))

]f−1

(d + h (N(Q))

rA

),

where N(Q) = v−1(m (Q− q)),

and the corresponding values x, y given by

x = f−1(

d + h(N(Q))rA

),

y = rAm(Q− q)/Qd + h(N(Q)) x,

where for notational convenience A = min{

1, Qθ

}and A = θ

Q A = min{θQ , 1}

.


Applications to Stoichiometric Producer-Grazer Models for Aquaculture Linear Functional Response







f (x) = ax

If the functional response f is the linear function f (x) = ax, the modeldynamics are greatly simplified and less interesting.

Theorem: E1 LAS = GAS

If f (x) = ax then grazer-only extinction E1 is globally asymptotically stable ifand only if

raT < dθ,

which is the same as the necessary and sufficient condition for LAS when f isdefined only generally.



f (x) = ax, cont’d

For the general f not defined explicitly, it is not feasible to find explicitconditions for the existence of any coexistence equilibria E2. However, withf (x) = ax:

Theorem: E2 existence and uniqueness

If f (x) = ax then the coexistence equilibrium E2 exists if and only if E1 isunstable, i.e.,

raT > dθ.

Further, there is only one unique E2.




0.5 1 1.50

0.5

1

1.5

2

2.5

3

←dθ=qrf(T/q)

y (m

g C

/L)

d (days−1)

Figure 5: y

0.5 1 1.50

2

4

6

8

10

12

14

←dθ=qrf(T/q)

x (m

g C

/L)

d (days−1)

Figure 6: x

Bifurcation on d for the model with linear functional response f (x) = ax.




0.5 1 1.50

0.1

0.2

0.3

0.4

←dθ=qrf(T/q)

Q (

mg

N/m

g C

)

d (days−1)

Figure 5: Q

0.5 1 1.50

2

4

6

8

10

12

14

←dθ=qrf(T/q)

x (m

g C

/L)

d (days−1)

Figure 6: x

Bifurcation on d for the model with linear functional response f (x) = ax.


Applications to Stoichiometric Producer-Grazer Models for Aquaculture Holling Type II Functional Response







f (x) = ax/(x + s)

If the functional response f is the Holling type II function f (x) = ax/(x + s), themodel dynamics are more complicated.

Differences from linear f

LAS and GAS of E1 are not equivalent.Multiple E2 can coexist.E2 can exist even if E1 is LAS.Bistability: both LAS E1 and either LAS E2 or a stable periodic orbit aboutE2 can exist simultaneously.



Holling Type II

0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

dθ / [rqf(T/q)]

y (m

g C

/L)

LASunstable

Figure 7: y

0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

dθ / [rqf(T/q)]

x (m

g C

/L)

LASunstable

Figure 8: x

Bifurcation on d for the model with Holling type II functional responsef (x) = ax/(x + s).



Holling Type II, cont’d

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

X

Y

←T=qX+θY

E2 (unstable)

E2 (LAS)

E1 (LAS)

E0 (unstable)

Figure 9: E1 and an E2 areLAS

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

X

Y

←T=qX+θY

E2 (unstable)

E1 (LAS)

E0 (unstable)

Figure 10: E1 is LAS; stableperiodic orbit about an E2

Orbits projected into the xy-plane illustrating bistability.



Holling Type II, cont’d

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

X

Y

←T=qX+θY

E2 (unstable)

E1 (LAS)

E0 (unstable)

Figure 9: E1 is LAS; stableperiodic orbit about an E2

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

X

Y

←T=qX+θY

E1 (LAS)

E0 (unstable)

Figure 10: E1 is GAS; no E2

exist.

Orbits projected into the xy-plane illustrating bistability.


Prostate Cancer and Androgen Deprivation Therapy

1 Introduction






Application to PCA

Applications to prostate cancer?

What if we consider androgen (testosterone) as a limiting nutrient?

Portz, T., Y. Kuang and J. D. Nagy, “A clinical data validated mathematical model ofprostate cancer growth under intermittent androgen suppression therapy”, AIPAdvances 2, 1, 011002 (2012).



Portz et al. (2012)

Xi: prostate cancer cells (cells×109),Qi: intracellular androgen concentration (nM),P: serum PSA concentration (ng/mL),Ts: serum testosterone concentration (nM).

X′1 = µm

(1− q1

Q1

)X1 − c1

(K1)n

(Q1)n + (K1)n X1 + c2(Q2)n

(Q2)n + (K2)n X2,

X′2 = µm

(1− q2

Q2

)X2 + c1

(K1)n

(Q1)n + (K1)n X1 − c2(Q2)n

(Q2)n + (K2)n X2,

Q′i = vmqm − Qi

qm − qi

Ts

Ts + vh− µm (Qi − qi)− bQi, i = 1, 2,

P′ = σ0 (X1 + X2) + σ1X1(Q1)m

(Q1)m + (ρ1)m + σ2X2(Q2)m

(Q2)m + (ρ2)m − δP.



It works!

Ex. result from Portz et al. (2012). Patient data from Akakura et al. (1993).

Akakura, K., N. Bruchovsky, S. L. Goldenberg, P. S. Rennie, A. R. Buckley, and L. D. Sullivan.“Effects of intermittent androgen suppression on androgen-dependent tumors. Apoptosis andserum prostate-specific antigen.” Cancer 71, 9 (1993): 2782–2790.



It works!

It works!But why, and how should it be interpreted?

X′1 = µm

(1− q1

Q1

)X1 − c1

(K1)n

(Q1)n + (K1)n X1 + c2(Q2)n

(Q2)n + (K2)n X2,

X′2 = µm

(1− q2

Q2

)X2 + c1

(K1)n

(Q1)n + (K1)n X1 − c2(Q2)n

(Q2)n + (K2)n X2,

Q′i = vmqm − Qi

qm − qi

Ts

Ts + vh− µm (Qi − qi)− bQi, i = 1, 2,

P′ = σ0 (X1 + X2) + σ1X1(Q1)m

(Q1)m + (ρ1)m + σ2X2(Q2)m

(Q2)m + (ρ2)m − δP.



It works!

ProblemsInterpretation of Q

So-called uptakeUse of Droop modelMechanism of treatment resistance“Mutation rate”


Prostate Cancer and Androgen Deprivation Therapy Mechanistic derivation







Return to basics

Modeling AR kinetics

Testosterone exchange between serum and prostate is proportional tothe blood flow rate to the prostate and the concentration gradient.Prostatic testosterone is uniformly distributed amongst the prostate cells.Free testosterone is enzymatically converted to DHT by 5α-reductase.Free testosterone and DHT bind to free AR in the cytoplasm by secondorder reaction kinetics.Free AR, T, and DHT degrade by first order kinetics.A fixed total AR concentration, Rt, is maintained at homeostasis.



AR kinetics model (Eikenberry et al. 2010)

CT(t): T:AR complex concentration (nM),CD(t): DHT:AR complex concentration (nM),R(t): intracellular free AR concentration (nM),D(t): intracellular free DHT concentration (nM),T(t): intracellular free T concentration (nM).

C′T = kTa TR− kT

d CT ,

C′D = kDa DR− kD

d CD,

R′ = λ− kTa TR + kT

d CT − kDa DR + kD

d CD − βRR,

D′ = αkcatT

T + KM− kD

a DR + kDd CD − βDD,

T ′ = K(Ts − T)− kTa TR + kT

d CT − αkcatT

T + KM− βTT.

Eikenberry, S. E., J. D. Nagy, and Y. Kuang. “The evolutionary impact of androgen levels onprostate cancer in a multi-scale mathematical model.” Biology Direct 5, 1 (2010): 1–28.



Population dynamics

Next step

Need to translate intracellular AR dynamics to population level: Q

Issues with Droop model

Usage and meaning not entirely clear.Instead, let’s use hill functions for the growth rate and newandrogen-dependent apoptosis rate.



Proliferation and apoptosis rates

“Cell quota”

Let Q be the sum of intracellular AR:T and AR:DHT complexes, CT + CD

(Eikenberry et al. 2010).

X′ = µX − δX,

P′ = σQp

Qp + (qσ)p X − βPP,

where

Q = CT + CD,

µ(Q) = µmQm

Qm + (qµ)m , δ(Q) = δm(qδ)n

Qn + (qδ)n + δ0.



Yikes!

For each subpopulation (not including the “mutation rates”):

X′ = µmQm

Qm + (qµ)m X −(δm

(qδ)n

Qn + (qδ)n + δ0

)X,

C′T = kTa TR− kT

d CT − µCT ,

C′D = kDa DR− kD

d CD − µCD,

R′ = λ− kTa TR + kT

d CT − kDa DR + kD

d CD − βRR− µR,

D′ = αkcatT

T + KM− kD

a DR + kDd CD − βDD− µD,

T ′ = K(Ts − T)− αkcatT

T + KM− βTT − µT,(

P′ = σQp

Qp + (qσ)p X + σ0X − βPP)


Prostate Cancer and Androgen Deprivation Therapy Uptake







Closer look at uptake


Is the model from phycology better after all?

Not necessarily

Possible to mechanistically derive the “uptake” function from the ARkinetics model.Doing so results in model that better fits data.




Quasi steady state

Intracellular model works on faster time scale than population dynamics. Wecan simplify the model using quasi steady states.

D = 1βD

aTT + s

,

T = 12

(vTs − αm/h− s) + 12

[(vTs − αm/h− αk)2 + 4vTsαk

]1/2.




Rewrite expression for T:

T = 12

(vTs − αm/h − αk) + 12

[(vTs + αm/h + αk)2 − 4vTsαm/h

]1/2,

= 12

[(vTs − αm/h − αk) + (vTs + αm/h + αk)

(1 − 4vTsαm/h

(vTs + αm/h + αk)2

)1/2],

(Note: The relation4vTsαm/h

(vTs + αm/h + αk)2 < 1 is already established by the fact that the

expression under the radical in previous slide is nonnegative.)




Taylor expansion:

T(n) = vTs − vTsαm/hvTs + αm/h + αk

− (vTsαm/h)2

[vTs + αm/h + αk]3 −

. . .− (2n)!2(1 − 2n)(n!)2

(vTsαm/h)n

[vTs + αm/h + αk]2n−1 .




The error for T(n) is 0 if Ts = 0 and is an increasing function of Ts. Error isapproximated by

‖T − T(n)‖ ≈ (vTs + αm/h + αk)2

(2n + 2)!(1 + 2n)((n + 1)!)2

(vTsαm/h)n+1

[vTs + αm/h + αk]2n+1

≤ 2vTsαm/hvTs + αm/h + αk

[4vTsαm/h

(vTs + αm/h + αk)2

]n

<2vTsαm/h

vTs + αm/h + αk4−n

<max {αm/h, vTs}

αk4−n. (6)

Therefore T(1) and T(2) are good approximations for T if αm/h is sufficientlysmall or αk is sufficiently large. Since v < 1 and Ts < αk for applications to ratand human, the condition αm/h is more pertinent.




Approximations:

T(1) = vTs D(1)I = αm

βD

vTs

vTs + αk

T(2) = vTsvTs + αk

vTs + αk + αm/hD(2)

I = αm

βD

vTs

vTs + αk

(vTs + αk)2

(vTs + αk)2 + αkαm/h.

Uptake:

V(Ts) = (Rt − Q)(

kTa T(Ts) + kD

a D(Ts))

V(1)(Ts) = vTs

[kT

a + kDaαm

βD

1vTs + αk

]V(2)(Ts) = vTs

vTs + αk

vTs + αk + αm/h

[kT

a + kDaαm

βD

vTs + αk + αm/h(vTs + αk)2 + αkαm/h

]


Prostate Cancer and Androgen Deprivation Therapy Single population model






Prostate Cancer and Androgen Deprivation Therapy Single population model

Simplified Model

We now have a model with the same structure of Portz et al. (2012):

X′ = µX − δm(qδ)n

Qn + (qδ)n X − δ0X,

Q′ = (Rt − Q)(kTa T + kD

a D)−((1− fD)kT

d − fDkDd

)Q− µQ,

P′ = σQp

Qp + (qσ)p X + σ0X − βPP,

where

T = vTsvTs + αk

vTs + αk + αm/h,

D = αm

βD

vTs

vTs + αk

(vTs + αk)2

(vTs + αk)2 + αkαm/h.

µ(Q) = µmQm

Qm + (qµ)m .


Prostate Cancer and Androgen Deprivation Therapy Results







Two Subpopulations: Cases 1 to 4

200 400 600 800 1000 1200 14000

2

4

6

8

10

12

14

16

18

20

days

Ser

um P

SA

(ng

/mL)

and

Cel

ls (

mill

ions

)

PSA data

PSA model

CS cells

CR cells

0 200 400 600 8000

10

20

30

40

50

60

days

Ser

um P

SA

(ng

/mL)

and

Cel

ls (

mill

ions

)

0 200 400 600 800 10000

10

20

30

40

50

60

70

days

Ser

um P

SA

(ng

/mL)

and

Cel

ls (

mill

ions

)

200 400 600 800 10000

5

10

15

20

25

30

35

days

Ser

um P

SA

(ng

/mL)

and

Cel

ls (

mill

ions

)

Model with two cell subpopulations fitted to patient data from Akakura et al. (1993).A. Packer Cell Quota Based Models + Applications : — 67 / 72


Two Subpopulations: Cases 5 to 7

0 100 200 300 400 500 6000

20

40

60

80

100

120

days

Ser

um P

SA

(ng

/mL)

and

Cel

ls (

mill

ions

)

0 100 200 300 400 500 6000

10

20

30

40

50

60

days

Ser

um P

SA

(ng

/mL)

and

Cel

ls (

mill

ions

)

200 400 600 800 10000

2

4

6

8

10

12

14

16

days

Ser

um P

SA

(ng

/mL)

and

Cel

ls (

mill

ions

)

Model with two cell subpopulations fitted to patient data from Akakura et al. (1993).A. Packer Cell Quota Based Models + Applications : — 68 / 72


One Population: Cases 1, 3, 4

200 400 600 800 1000 1200 14000

20

40

60

80

days

Ser

um P

SA

(ng

/mL)

and

Cel

ls (

mill

ions

)

PSA dataPSA modelcells

0 200 400 600 800 10000

10

20

30

40

50

60

70

days

Ser

um P

SA

(ng

/mL)

and

Cel

ls (

mill

ions

)

200 400 600 800 10000

10

20

30

40

days

Ser

um P

SA

(ng

/mL)

and

Cel

ls (

mill

ions

)

Model with one cell population fitted to patient data from Akakura et al. (1993).A. Packer Cell Quota Based Models + Applications : — 69 / 72


One Populations: Cases 5 to 7

0 200 400 6000

20

40

60

80

100

120

days

Ser

um P

SA

(ng

/mL)

and

Cel

ls (

mill

ions

)

0 200 400 6000

10

20

30

40

50

60

days

Ser

um P

SA

(ng

/mL)

and

Cel

ls (

mill

ions

)

200 400 600 800 10000

5

10

15

20

25

days

Ser

um P

SA

(ng

/mL)

and

Cel

ls (

mill

ions

)

Model with one cell population fitted to patient data from Akakura et al. (1993).A. Packer Cell Quota Based Models + Applications : — 70 / 72

Prostate Cancer and Androgen Deprivation Therapy Future Work






Prostate Cancer and Androgen Deprivation Therapy Future Work

Future Work

PredictionsCurrent models are poor at making predictions.Can we formulate a mechanistic model that accurately predicts treatmentoutcomes?

Stochastic methodsBayesian inference.Forecasting.


Cell Quota Based Population Models and their Applicationskuang/paper/AaronPHD.pdf · 2014-11-18 ·...

Documents

Transcript of Cell Quota Based Population Models and their Applicationskuang/paper/AaronPHD.pdf · 2014-11-18 ·...