Prostate Antigen Kinetics

Alan White, President 
Armillary Equations, LLC

(RowDefault[, (Spacer1), , (Spacer1), ])  #WolframTechConf

ABSTRACT

Differential equations, solved to the task of graphing the change in concentration of the Prostate Specific Antigen in prostate cancer patients. Note: most equations, original to the writings of Ernest Rutherford.  

         Baron Rutherford’s motto: Primordia Quaerere Rerum

                             To seek the first principles of things
                             
   Ernest Rutherford, First Baron Rutherford of Nelson, OM, FRS, 
                HonFRSE (30 August 1871 – 19 October 1937)

Differential Equations versus Linear Approximations

Logistic Equation versus (1) PSA Velocity (2) PSA Doubling Time

Correlating Histology with Differential Equations

Classroom Simulations (Logistic Manipulate)

Classroom Simulations (Table / Test)

Post Operative Patients

EQUATION A: Mono-exponential Depletion, ‘Washout’

Post Op Patients, A. Mono-exponential Depletion

RutherfordE1 = N0 x Exp[-(k x t )]
N0 = Initial Concentration, k = Time Coefficient

Logistic Equation (Historical Background)

Logistic Equation (Historical Background)

Post Operative Logistic Depletion

PSApstOp = N0 - (N0 x (I0/I0 + (100.0 - I0) x (Exp[-(k x t)])))) 
N0 = Initial Concentration, I0 = Increment, k = Time Coefficient

Post Operative Mixture Distributions (Exponential / Logistic)

Post Operative Data Analysis (Exponential / Logistic)

(1) RutherfordE1 = N0 x Exp[-(k x t )] 
                   N0 = Initial Concentration, k = Time Coefficient
(2) PSApstOp = N0 - (N0 x (I0/I0 + (100.0 - I0) x (Exp[-(k x t)])))) 
   N0 = Initial Concentration, I0 = Increment, k = Time Coefficient

Pharmacokinetic Distribution Models, PSA

Pharmacokinetic Distribution Models, PSA

Exponential Depletion? Logistic Depletion? PSMA Molecule mutations 
    by analogy with the CCR5 Co-Receptor molecule mutations?

Radiotherapy Dynamics (External Beam)

EQUATION B: Biphasic Tumor Cell Ablation

Radiotherapy Dynamics (External Beam)

RutherfordE2:= N0+((Pf x k1)/(k2-k1) x ((Exp[-(k1 x (t-Pk))])-(Exp[-(k2 x (t-Pk))])));
        N0 = Initial Concentration, Pf = Power Factor, Pk = Peak Locator, k1, k2 = Time Coefficients

Radiotherapy Dynamics (External Beam)

Radiotherapy Dynamics (Brachytherapy)

EQUATION C: Biphasic Tumor Cell Ablation Leading to a Plateau

Radiotherapy Dynamics (Brachytherapy)

RutherfordE3:=N0 + (Pf x k1 x k2) x ((Exp[-(k1 x (t/Pk))])/((k2-k1) x (k3-k1)) + (Exp[-(k2 x (t/Pk))])/
((k1-k2) x (k3-k2)) + (Exp[-(k3 x (t/Pk))])/((k1-k3) x (k2-k3))); N0 = Initial Concentration, 
Pf = Power Factor, Pk = Peak Locator, k1, k2, k3 = Time Coefficients (Red Dots, Blue Dots)

E3Rjthfrd:= N0 + ((k1 x Pf)/(k2 - k1)) x ((Exp[-(k1 x (t - Pk))]) - (Exp[-(k2 x (t - Pk))])); (Purple Dots)
Patient to Patient Comparisons: Graph each trajectory from peak to nadir against the mono-exponential and logistic 
depletion equations, then bell curve the time coefficients.

Logistic Tumor Growth: Stable? Unstable?

Stable Tumor Growth: inflection point passes through the center of the bounding rectangle. Unstable Growth: (a) inflection point located above (growth accelerated by copy number mutations?) (b) inflection point located below (growth decelerated by mutations?) the center of the bounding rectangle.

Post-Radiotherapy: Harmless Bounce? Biochemical Recurrence?

Bounce: usually at 17 months | Failure: usually at 41 months

Post-Radiotherapy: Harmless Bounce? Biochemical Recurrence?

Biochemical Recurrence, Differential Equations? Linear Approximations?

Radiotherapy and Quantum Mechanics

Planck M (1901) Über das Gesetz der Energieverteilung im Normalspektrum. Annalen der Physik. 309 (3) Pages: 553-563. doi:10.1002/andp.19013090310. Translated: Ando K On the Law of Distribution of Energy in the Normal Spectrum (PDF)

Radiotherapy and Quantum Mechanics                                             
                      (Healthy cells are radio resistant)

Radiotherapy and Quantum Mechanics

Gleason, Donald F; Mellinger George T (Feb 2002) “Prediction of prognosis for prostatic adenocarcinoma by combined histological grading and clinical staging. 1974”. Journal of Urology. United States. 167 (2 Pt 2): 953–8. doi:10.1016/S0022-5347(02)80309-3 (First Publication: Cancer Chemotherapy Reports, 1966)

Planck M (1901) Über das Gesetz der Energieverteilung im Normalspektrum. Annalen der Physik. 309 (3) Pages: 553-563. doi:10.1002/andp.19013090310. Translated: Ando K On the Law of Distribution of Energy in the Normal Spectrum (PDF)

Radiotherapy and Quantum Mechanics

Radiotherapy and Quantum Mechanics

Well oxygenated nutritionally rich cells | Hypoxic nutritionally poor cells

Radiotherapy and Quantum Mechanics

The Limit Concept (defined). Any discrete process, given a sufficient number of permutables and trials will take on a smooth continuous appearance to the observer. Thanks to Professor Stephen Stigler, PhD.

Radiotherapy, Quantum Mechanics, Black Body Radiation

Radiotherapy, Quantum Mechanics, Black Body Radiation

“According to Section 139, the “distribution density” w of the oscillators in the state of statistical equilibrium changes abruptly from one region element to another, while, according to Section 138, the distribution within a single region is uniform.” Planck M (1914) The Theory of Heat Radiation. Page: 271
 “According to the quantum theory, a molecule of a definite kind may, aside from its orientation and its translational motion, be in one of only a discrete set of states Z1, Z2, . . . Zn . . . whose (internal) energies are ε1, ε2, . . . εn . . . .  If the molecules of this kind belong to a gas of temperature T, then the relative abundance Wn of the state Zn is given by the statistical mechanical canonical partition function for states Wn = pne-^εn/kT” (5)” Einstein A (1917) Page: 3

Radiotherapy and Quantum Mechanics

Popular applications of the Logistic (Verhulst) Equation.

Radiotherapy and Quantum Mechanics

Optimal time between doses?

Radiotherapy and Quantum Mechanics

Optimal time between doses? M Tu W Th F Sa Su M

Radiotherapy, Quantum Mechanics, Catastrophe Theory

H-RT Should be the Standard of Care for Men with Low Risk Prostate Cancer. Primary outcomes demonstrated no difference in progression and survival between treatment arms. Men on the first arm received C-RT at 73.8 Gy in 41 fractions (1.8 Gy / dose). Men on the second received H-RT at 70 Gy in 28 fractions (2.5 Gy / dose).

Radiotherapy and Quantum Mechanics

1. Radiosensitivity 2. Redistribution 3. Repair 4. Reoxygenation 5. Repopulation

Prostate Specific Antigen Kinetics, a Calculus Text

Sir Isaac Newton (25 December 1642 – 20 March 1726/27 ) 
“Go see Mister De Moivre. He knows more of these things than do I.”

Wilhelm Ostwald (2 September 1853 - 4 April 1932)
“But here I must conclude, for I have ventured to intrude on a field where I have not secured my own right of entry by personal work. I see among my audience men who are possessed of an incomparably more minute and comprehensive knowledge of these new realms of science than I. I must ask you, therefore, to take these suggestions in the same spirit as that which Faraday took his own speculations. They are questions put to nature. If she says Yes, then we may follow the same path a little further. If she says No- well, then we must try another path.”

Wilhelm Ostwald. Elements and Compounds THE FARADAY LECTURE, delivered before the Fellows of the Chemical Society in the Theater of the Royal Institutes on Tuesday, April 19th, 1904.

- The End -