A Model of Natural Selection Predicts Treatment Resistance

Introduction Model Results Summary

A Model of Natural Selection PredictsTreatment Resistance in Prostate Cancer

John D. Nagy

Department of Life Science, Scottsdale Community CollegeSchool of Mathematical and Statistical Sciences, Arizona State University

Contributed Session 9: Cancer Treatment

SMB 2017Salt Lake City, UT, USA

19 July 2017

J. D. Nagy Evolution of Androgen Independent Prostate Cancer


Collaborators

Yang KuangSchool of Mathematical and StatisticalSciences, Arizona State University

Heiko EnderlingIntegrated Mathematical Oncology,H. Lee Moffitt Cancer Center

Students:Khoa HoWilliam BakerPaige MitchellJonathan TrautmanChandler GrantAlaina Daum


Introduction Model Results Summary Biology Clinical course Causes

Discovery of Androgen Dependence in Prostate Cancer

Charles Hodges

Huggins and Hodges (1941, Cancer Res.)—first evidencethat prostate cancer cells are hormone dependent.

Huggins awarded the Nobel Prize in Physiology orMedicine in 1966 for the discovery.

Shared the prize with Peyton Rous.



Androgen-dependent tumor growth in context

Tumor mass =Sum of

clonogenicstrains

ScalesTime units: daysSpatial units: cm

Healthy Prostate

Tumor

Ts (serum testosterone)

ScalesTime units: minsSpatial units: μm

Afferent blood

Serum PSA

Efferentblood

T (testosterone)

D (DHT)

5-α reductaseR (AR)

AR:androgencomplex

Proliferation

AR-dependentgene

expression



Intermittent androgen deprivation therapy

Data from: CanadianProspective Phase IITrial of IntermittentAndrogen Deprivation (Bruchovsky, Klotz, etal., early 2000s).

Enrolled participants allhad radiation refractoryprostate cancer

Shaded region: On Tx(Lupron + CPA)



Intermittent androgen deprivation therapy



Typical outcome, castration resistance



Ultimate cause of castration resistance

Research question

Why do prostate malignancies become hormone refractoryunder androgen ablation?

Isaacs and Coffey (1981) consider two competing hypotheses:

1 Plasticity (pseudo-adaptation)

2 Natural selection (true adaptation)



Elucidation of the hypotheses

PlasticityHypothesis

NaturalSelectionHypothesis


Introduction Model Results Summary Scheme Model

Modeling schematic

Direction of blood flowDirection of blood flow

Serum [Androgen] = Serum [Androgen] = yy00

InterstitialInterstitial [Androgen][Androgen]

= = yy11

α1

α1

α2

IntracelluarIntracelluar[Androgen] = [Androgen] = yy

22

Unbound [AR] = Unbound [AR] = zzuu

QQ = = zzuu + + zz

bb

Bound [AR] = Bound [AR] = zzbb

r1

r2



Model

Molecular dynamics:

y1 =c+ k

k(α1(y0 − y1) + α2(y2 − y1))− y1Φ(zb, ·),

y2 =c+ k

cα2(y1 − y2)− ηy2 − r1zuy2 + r2zb − y2Φ(zb, ·),

zu = P (zb, ·)− r1zuy2 + r2zb,

zb = Q− zu,Φ(zb, ·) = P (zb, ·)−M(zb, ·).

Tumor dynamics:

x = Φ(zb, ·)x= (P (zb, ·)−M(zb, ·))x= (P (zb, ·)−m(zb, ·)− ξ(Q))x.

P : Proliferation rate.m : “Natural” mortality rateξ : AR production cost.



Selection on reaction norm without treatment



Selection on reaction norm with treatment


Introduction Model Results Summary Evolution Patient forecasts

Invasion principle from adaptive dynamics theory

Suppose a rare mutant clone arises in an established, otherwisemonomorphic tumor. Let the resident clone’s mass be xr andthe mutant’s xm, and consider

z =xm

xr + xm,

and letΦi = Φ(zb(Qi, y0), Qi).

Thenz = z(1− z) (Φm − Φr) .

Invasion criterion

Φm < Φr ⇒ mutant cannot invade.

Φm > Φr ⇒ mutant can invade (but may go extinct due torandom forces).

Φm = Φr ⇒ mutant is evolutionarily neutral (drift).



Invasion principle gives the ESS reaction norm



Example: Good control, no resistance

Points: Patient data.

Black: Model solution.

Lt. blue: Simulations.

Dashed lines: ESS.

Curves: Canonical solution.

Lt. blue: Simulations



Example: Impending castration resistance

Right: Heavy line is castration resistance threshold. Upperdashed line is ESS on Tx.Prediction: This patient was a cycle or 2 away from castrationresistance at study endpoint.



Example: Realized castration resistance

The model correctly predicts timing of castration resistance,but. . .



What about our original patient, 47?

. . . sometimes fails to predict hormone refractory tumors.



Summary

Hypothesis

Natural selection for the AR set-point reaction norm ultimatelyexplains hormone refractory prostate cancer.

A multi-scale model of this idea performs well but notperfectly. In 30 patients, the model

correctly predicted normal exit 17 times (57%);correctly predicted castration resistance 5 times (17%);incorrectly predicted tumor control 1 time (3%);incorrectly predicted castration resistance 2 times (6%);made just an awful prediction 5 times (17%).

Why does it fail at times?

Change in AR set point is only one, although the mostcommon, mechanism of castration resistance.Most poor fits associated with unrelated adverse medicalevents.



Thanks and acknowledgements

Thanks to

Nicholas Bruchovsky for the trial data;

Jim Elser for insights;

David Ung, Kirsten Karr and Karl Lundin for early workon this project;

ASU and SCC for supporting the research;

SCC for funding travel to SMB;

you folks for listening.


Documents

A Model of Natural Selection Predicts Treatment Resistance