Predicting Long Term Response to Treatment for Prostate Cancer Based on Short Term Linear Regression...

Predicting Long Term Response to Treatment for Prostate Cancer

Based on Short Term Linear Regression

byDr. Deborah Weissman-Berman

PROGRAM50th Anniversary Celebration

of FSU’s Statistics Department

Predicting Long-Term Response to Treatment The prediction is made from a

Linear regression for short-term data Paired with a predictive convolution

integral – an ‘hereditary integral’' From continuum mechanics

Method broadens possibility for statistical methods in

Survival & hazard analysis

The Method Motivation

Prostate cancer is one of the most common forms of cancer in American males

Long-term predictions can aid in clinical treatment

Methodology Presents a time-dependent method For predicting long-term antigen-free outcomes

from brachytherapy localized treatment with iodine-125

Predicting Long-Term Response to Treatment (1) Prediction involves

Derive the ‘change point’ for F statistic From linear regression models

(2)Use the resulting equations For a pairing scheme From mechanics to statistics

(3)Derive value of shape parameter (4)To predict antigen-free survival

From an hereditary integral

Predicting Long-Term Response to Treatment

The data set: Data given by Joseph, et al.(2004)

Predicts K-M survival curves Relapse-free survival of 667 patients Treated by brachytherapy

Implantation of iodine-125 Localized treatment

Predicting Long-Term Response to Treatment

This methodology assumes: Initial portion of any treatment curve Considered linear This data gathered by the investigator

This portion of the curve can be defined

And modeled by Simple linear regression

Derivation of the Change Point

(1) Definition of a ‘change point’ A point in time where the character of

the regression changes The point at which there is A retardation effect of the response

At the value of this F statistic The corresponding time point Is the value input into the predictive

equations as


Value of the ‘change point’ Determined by

The most significant or least significant F statistic

For simple linear regression Models using;

ii exy 10


Figure (1) Initial portion of the curve assumed linear


(a) A first approximation The nested models approach To determine the F statistic

2ˆ

)/(

AHNHAHNH dfdfRSSRSS

F

Derivation of the Change Point General strategy

Start with the largest 8 week model Then a smaller model – for 6 weeks is

nested Results

F statistic continuously decreases

Results: F statistic continuously decreases

Not relevant to determine most or least significant F statistic


(b) A second approximation Pooled information across genes For small sample data from Wu (2005)

The matrix for gene expression data: Where the first n1 samples are the 1st

group The last n2 are from the second group

21,,...,1,,...,1, nnnnjmix ji


The comparison for gene i Is from a linear regression model:

Testing the difference by:

njyx jjji ,...,1,10

01


And:

Where F = t^2

21

221

211

21

1

1

2*

)()()ˆ(

ˆ

nnn

n

xxxx

xx

st

iijnjiijnj

ii

ei

221

211

221212

1)()(

))(/()2(

iijnjiijnj

ii

xxxx

xxnnnntF

Derivation of the Change Point Which yields:

Results: Using such a pooled estimate

From say groups 4 & 5 Yield continuously increasing

values of F statistic

SSE

SSRn

SSW

SSBn )2()2(

Derivation of the Change Point (c) Determining the most or least

significant F statistic Following the logical derivation of the F statistic, given by Wu (2005)

An F statistic is derived from: Describing the parameters of interest Deriving the t-test statistic Deriving the F statistic = t^2


The parameters:

SXX

SXY1̂ xy 10

ˆˆ

n

i ii

n

i iie yyn

xyn

s1

2

1

2101 )ˆ(

2

1)ˆˆ(

2

1)ˆ(

),(~ˆ1̂

211 N ),(~ˆ

0ˆ2

00 N


Then:

And:

21)(

))((ˆxx

yyxx

SXX

SXY

i

ii

n

i ii

iii

e xyn

xxyyxx

st

1

210

2

1

1

)ˆˆ(2

1

)(/))((

)ˆ(

ˆ


With:

After algebraic manipulation:

2

22

1

2

)(

))(()(

)(

xx

yyxxyy

SXX

SXYSYYRSS

i

iin

i i

222

10

2

2

22 ˆ

Re

ˆ

1/Re

)ˆˆ(2

1)(

)])(([

ˆ

1/)(

gMSgSS

xyn

xx

yyxx

tRSSSYY

F

iii

i

ii


Table 1 – Results of backward stepwise elimination method for ________________

Time/months

Bio free from failure

R^2 F statistic RSS MS

6 .985 .690 0.690 .00003 .00006

8 .970 .810 25.583 .00011 .0005

10 .950 .859 48.529 .0003 .0021

12 .920 .875 70.105 .0009 .0061

14 .890 .886 93.631 .0020 .0155

16 .880 .921 162.365 .0023 .0269

18 .855 .943 265.220 .0024 .0403

20 .840 .959 416.225 .0025 .0572

21 .840 .963 497.082 .0025 .0653

22 .840 .965 548.216 .0027 .0725

23 .840 .964 554.739 .0030 .0789

24 .840 .960 524.419 .0036 .0845

25 .840 .954 475.603 .0043 .0894

Derivation of the Change Point The time corresponding to the F

statistic At the change point Is used as the input to :

In the kernel of the time-dependent convolution integral:

And as:

1

0

0

),1(1

)(q

qe

qtJ t

)1(1

)( t

eE

tJ

The change, or relaxation point of the data

Derivation of the Change Point Graphic results scatter matrix for Prostate Cancer Data

Time

0

100

200

300

400

500

2 7 12 17 22

0 100 200 300 400 500

Fstat

2

7

12

17

22

R.2

0.65

0.75

0.85

0.95

0.65 0.75 0.85 0.95

Predicting Long-Term Response to Treatment domain

Figure (2) Predicted portion of the curve (23-100 months)

Mechanics to statistics

(2) Compare variable slopes

E

1/D is known as a compliance term. This term can be related to a function of time

1/G is also a form of a compliance term. This term will be related to a function of time in this analysis.

Mechanics: Statistics:

wx

DE

1

Gwx

wx 10

xw

Figure (3) mechanics compared to statistics slopes and compliance

0


The compliance term in statistics

Can be related the same way as in mechanics

Gwx

wx 10

DE

1


Then the function

Can be given as a function of time

Where

Gwx

wx 10

)(

1, tGtw

wxandxwbothoffunctionais 0


For the bivariate function – there are 2 equations:

To predict, we have:

xtw wtG

w )(

1, wxtwx tG

)(

1.

wxxi

txtwx w

w 1,

,xwx

twxtx ww

,

,

Derivation of shape parameter

(3) Weibull Distribution Parameters

Support

)(0 realscale)(0 realshapek

];0[ t

Derivation of shape parameter ‘k’

cdf of Weibull distribution

shape function for predictive equation - when evaluation of Weibull distribution for ‘k ‘ for least squares regression at equals ‘m’ (slope):

kte )/(1

mkte

xw

)/(1

1

Derivation of shape parameter

Solve for k

Prostate cancer data = 1.0942

/ln

ln

te

mmxw

k

k

Hereditary Integral The Kelvin model

A spring A dashpot, in parallel

Used in this integral This model – think muscular-

skeletal structure and blood To model human response

To treatment for disease.

)1( t

ec

Hereditary Integral

Hereditary integral

With initial discontinuity at t=0

tdtd

tdttJt

t

)()()( 12

12

tdtd

tdttJtjt

t

)(

)()()( 12

0

012

Hereditary Integral

The model for the hereditary integral:

Is embedded in a LaPlacian time step –then:

1

0

0

),1(1

)(q

qe

qtJ t )1(

1)(

t

eE

tJ

t

tdtd

dttjtJte

0

0 )()()(

Hereditary Integral

The final result after integrating by parts and the use of a LaPlace transform is:

Finally;

)()1(1

)( 1

0

)(

01

1

01

10

tJtdeqt

eqt

tt tttt

)(, tJww xtx

0

Hereditary Integral (3) Results are used for predictive

model Note that here Then for

And for upper bound asymptote

wxtwx tG

)(

1.

wx

wxx

asmpttxtwx w

w )(

)(,,

wxxi

txtwx w

w 1,

,

Hereditary Integral

Exponential distribution of the survival function is

Where the kernel of this predictive function shows precedence in survival analysis

)1()( tetF

1

0

0

),1(1

)(q

qe

qtJ t

Results

Table 2 – Response Summary for Gleason score = 7

Time in months

change point Wx,t Wx,t(k) Ratio factor (wx,t(k))

23 23 36.367 39.793 --- .842 (asmp)

26 23 33.968 37,168 .979 (26/25) .786

30 23 31.565 34.538 .984 (30/29) .771

40 23 27.899 30.527 .991 (40/39) .754

50 23 25.829 28.262 .990 (50/49) .747

60 23 24.817 27.155 .996 (60/59) .744

70 23 24.150 26.425 .998 (70/69) .742

80 23 23.736 25.972 .999 (80/79) .742

90 23 23.469 25.670 .978 (90/89) .726

Results

Comparison of Tested and Predicted Prostate Data

tau = 23 months (most/least) inlinear regr. data) Correlation to

Gleason score = 7

Comments

Predictive equation set Independent of number of subjects

‘n’ Therefore can be used for single subject design and For clinical comparative interpretation Of individual response to RCT data

Results for Obesitytau = 6 weeks (most/least) inlinear regr. data)

+ corr. to placebo corr. to cont. phen. corr. to inter. phen.

Comparison of Tested and Predicted Weight Loss

Results-falls in elders

Correlation to controlsCorrelation to patients

Comparison of Tested and Predicted falls in elders

9070

(Reproduced by permission BMJ Publishing Group Ltd.)

Discussion Method is useful pairing

Of statistical regression line data With mathematical (hereditary)

convolution integral For prediction of antigen-free

survival In prostate cancer

In obesity weight loss In reduction of falls in elders

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