Predicting Long Term Response to Treatment for Prostate Cancer Based on Short Term Linear Regression by Dr. Deborah Weissman-Berman PROGRAM 50 th 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

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;

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

Derivation of the Change Point (a) A first approximation The nested models approach To determine the F statistic

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

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 1 st group The last n2 are from the second group

Derivation of the Change Point The comparison for gene i Is from a linear regression model: Testing the difference by:

Derivation of the Change Point And: Where F = t^2

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

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

Derivation of the Change Point The parameters:

Derivation of the Change Point Then: And:

Derivation of the Change Point With: After algebraic manipulation:

Derivation of the Change Point Table 1 – Results of backward stepwise elimination method for ________________ Time/month s Bio free from failure R^2F statisticRSSMS

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

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

Predicting Long-Term Response to Treatment domain Figure (2) Predicted portion of the curve ( 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: Figure (3) mechanics compared to statistics slopes and compliance

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

Mechanics to statistics Then the function Can be given as a function of time Where

Mechanics to statistics For the bivariate function – there are 2 equations: To predict, we have:

Derivation of shape parameter (3) Weibull Distribution Parameters Support

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):

Derivation of shape parameter Solve for k Prostate cancer data =

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.

Hereditary Integral Hereditary integral With initial discontinuity at t=0

Hereditary Integral The model for the hereditary integral: Is embedded in a LaPlacian time step – then:

Hereditary Integral The final result after integrating by parts and the use of a LaPlace transform is: Finally; 0

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

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

Results Table 2 – Response Summary for Gleason score = 7 Time in months change point Wx,tWx,t(k)Ratio factor (wx,t(k)) (asmp) , (26/25) (30/29) (40/39) (50/49) (60/59) (70/69) (80/79) (90/89).726

Results Comparison of Tested and Predicted Prostate Data tau = 23 months (most/least) in linear 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 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

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.)

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