Predicting Long Term Response to Treatment for Prostate Cancer Based on Short Term Linear Regression by Dr. Deborah Weissman-Berman

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Predicting Long Term Response to Treatment for
Prostate Cancer Based on Short Term Linear
Regression byDr. Deborah Weissman-Berman

• PROGRAM
• 50th Anniversary Celebration of
• FSUs Statistics Department

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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

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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

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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

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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

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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

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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

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Derivation of the Change Point

• Value of the change point
• Determined by
• The most significant or least significant F
  statistic
• For simple linear regression
• Models using

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Derivation of the Change Point
Figure (1) Initial portion of the curve assumed
linear
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Derivation of the Change Point

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

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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

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Derivation of the Change Point

• (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

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Derivation of the Change Point

• The comparison for gene i
• Is from a linear regression model
• Testing the difference by

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Derivation of the Change Point

• And
• Where F t2

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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

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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 t2

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Derivation of the Change Point

• The parameters

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Derivation of the Change Point

• Then
• And

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Derivation of the Change Point

• With
• After algebraic manipulation

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Derivation of the Change Point
Table 1 Results of backward stepwise elimination method for ________________ Table 1 Results of backward stepwise elimination method for ________________ Table 1 Results of backward stepwise elimination method for ________________ Table 1 Results of backward stepwise elimination method for ________________ Table 1 Results of backward stepwise elimination method for ________________ Table 1 Results of backward stepwise elimination method for ________________
Time/months Bio free from failure R2 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
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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

The change, or relaxation point of the data
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Derivation of the Change Point

• Graphic results scatter matrix for Prostate
• Cancer Data

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Predicting Long-Term Response to Treatment domain
Figure (2) Predicted portion of the curve (23-100
months)
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Mechanics to statistics

• (2) Compare variable slopes

Figure (3) mechanics compared to statistics
slopes and compliance
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Mechanics to statistics

• The compliance term in statistics
• Can be related the same way as in mechanics

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Mechanics to statistics

• Then the function
• Can be given as a function of time
• Where

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Mechanics to statistics

• For the bivariate function there are 2
  equations
• To predict, we have

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Derivation of shape parameter

• (3) Weibull Distribution
• Parameters
• Support

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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)

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Derivation of shape parameter

• Solve for k
• Prostate cancer data
• 1.0942

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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.

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Hereditary Integral

• Hereditary integral
• With initial discontinuity at t0

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Hereditary Integral

• The model for the hereditary integral
• Is embedded in a LaPlacian time step then

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Hereditary Integral

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

0
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Hereditary Integral

• (3) Results are used for predictive model
• Note that here
• Then for
• And for upper bound asymptote

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Hereditary Integral

• Exponential distribution of the survival function
  is
• Where the kernel of this predictive function
  shows precedence in survival analysis

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Results
Table 2 Response Summary for Gleason score 7 Table 2 Response Summary for Gleason score 7 Table 2 Response Summary for Gleason score 7 Table 2 Response Summary for Gleason score 7 Table 2 Response Summary for Gleason score 7 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
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Results
tau 23 months (most/least) in linear regr.
data)
Correlation to Gleason score 7
Comparison of Tested and Predicted Prostate Data
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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

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Results for Obesity
tau 6 weeks (most/least) in linear regr. data)
corr. to placebo corr. to cont. phen.
corr. to inter. phen.
Comparison of Tested and Predicted Weight Loss
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Results-falls in elders
Correlation to controls Correlation to patients
Comparison of Tested and Predicted falls in elders
(Reproduced by permission BMJ Publishing Group
Ltd.)
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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