Abstract Objectives To provide a mathematical background for understanding the phenomenon of analyte hemodilution using a kinetic analysis. Methods The first assumption for this analysis is that change in concentration of any analyte, such as prostate-specific antigen (PSA), is due to the flux of the analyte from an organ into the blood minus its flux from the blood. What results is a relatively simple differential equation that emphasizes the importance of plasma volume, organ mass, and two rate constants. Results The analyses demonstrate how serum PSA can be affected by plasma volume as well as body mass and how hemodilution due to obesity can be at least partly corrected for by expressing PSA in units of total mass or total mass density. Conclusions At a time when obesity is prevalent, expressing analytes in units of total mass may make them relate more closely to disease status and prognosis. Hemodilution, Obesity, Plasma volume, Kinetics, PSA, PSA mass When an organ releases an analyte into the blood, the principle of conservation of mass implies that the rate of change in concentration of the analyte must reflect the mass of analyte entering the blood as well as the loss of analyte from the bloodstream. If y symbolizes the concentration of analyte, then the foregoing implies the following kinetic equation: dy/dt = α * MV β * y (1) Here, M symbolizes the mass of the organ (g), V symbolizes plasma volume (mL), and α reflects the amount of analyte released per unit time and per unit weight of the organ (ng/(day × g)). β provides the rate for loss of the analyte from the blood (1/days), regardless of whether that loss is physiologic or chemical. Organs may release analytes when they are damaged, as with cardiac markers during acute myocardial infarction, or when they harbor a tumor, as with serum prostate-specific antigen (PSA) and prostate cancer. In the first example, α reflects the size of the infarct.1 In the second, α reflects the size and grade of the tumor as well as any benign tissue capable of releasing PSA.2 Inspection of equation (1) reveals the importance of V. For example, if two patients have different values of V, then the release of the same amount of analyte (ie, α * M) yields a higher concentration in the one with the lower V. This phenomenon has been called the hemodilution effect.3 Although V is not routinely measured in patients, V is reflected by body size.4,5 Thus, at a time when obesity has become common,6 the levels of any analyte may be confounded by hemodilution. In what follows, I use the example of serum PSA to examine the relationships between concentration, PSA mass, V, M, α, and β. Materials and Methods Model for Plasma Volume Data providing plasma volume (V) determined by the Evans blue dye dilution method were obtained from two prior studies4,5 to yield a total of 172 subjects, and regression analysis was performed to relate V to height, weight, sex, and hematocrit (Hct). The mean value of V in the men of these data was 3,063 mL, and other details of the study population are given in Table 1 Table 1 Data Used to Relate Plasma Volume to Height and Weighta Variable Mean Median Range Height, cm 168 167 143-190 Weight, kg 67.4 65.4 43-132 Hematocrit, % 43 43 34-53 Plasma volume, cc 2,693 2,631 1,750-4,070 Variable Mean Median Range Height, cm 168 167 143-190 Weight, kg 67.4 65.4 43-132 Hematocrit, % 43 43 34-53 Plasma volume, cc 2,693 2,631 1,750-4,070 a Data came from Gibson and Evans4 and Retzlaff et al.5 Fifty-two percent of the study persons were female. View Large Kinetic Model For this analysis, y symbolizes the concentration of PSA in the plasma, dy/dt symbolizes PSA velocity, and M symbolizes prostate weight. Although equation (1) is a first-order differential equation with a solution (see supplemental material, which can be found at American Journal of Clinical Pathology online), the solution is too complex for practical use. Nevertheless, a simplified form of equation (1) can be obtained. For example, many have observed that serial values of PSA follow an exponential pattern7,8 (see Humphrey and Vollmer7 for an early review), and this exponential pattern implies that at any time prior to diagnosis and treatment, the PSA velocity is proportional to the PSA concentration. Mathematically, this statement becomes dy/dt = ρ * y (2) Here, ρ has been termed a relative velocity,9 because it gives the PSA velocity relative to the value of PSA at the same time and can be written as ρ = PSA Velocity / PSA (3) ρ has been shown to be prognostic and to relate to tumor stage and growth rate as well as to outcomes.7,9 Furthermore, if PSA measurements at two different times, t1 and t2, are symbolized as y1 and y2, then ρ can be estimated as ρ = ln(y2/y1) / (t2−t1) (4) with ln symbolizing the natural logarithm.9 Equation (4) implies that ρ equals .693/(PSA doubling time). Finally, substituting equation (2) into equation (1) and solving for y yields y =α * MV * (ρ + β) (5) This result allows us to demonstrate the expected relationships between serum PSA and V, body weight, and M. Results Relationship of Plasma Volume to Height, Weight, Sex, and Hct Table 2 shows the results of a linear regression analysis relating V to height (cm), weight (kg), female sex, and Hct (data obtained from Gibson and Evans4 and Retzlaff et al5). Clearly, all four variables were closely related to V. Figure 1 shows a plot of observed values of plasma volume on the x-axis vs those predicted by the regression model on the y-axis, and the line shows where perfect agreement would occur. The plot demonstrates a significant amount of noise in the regression relationship even though the model explained 99% of the variance. Table 2 Linear Regression Analysis of Plasma Volumea Variable Coefficient P Value Height, cm 19.93 ~0 Weight, kg 16.1 ~0 Sex –409 ~0 Hematocrit, % –35.8 ~0 Variable Coefficient P Value Height, cm 19.93 ~0 Weight, kg 16.1 ~0 Sex –409 ~0 Hematocrit, % –35.8 ~0 a Sex was coded as 1 if female and otherwise as 0. P values of ~0 indicate that the P value was so low that the S-PLUS software could not discriminate it from zero. View Large Figure 1 View largeDownload slide Plot of the predicted plasma volume (Vp) obtained from the linear regression model of Table 1 vs the plasma volume observed in the 172 patients used for the analysis. Figure 1 View largeDownload slide Plot of the predicted plasma volume (Vp) obtained from the linear regression model of Table 1 vs the plasma volume observed in the 172 patients used for the analysis. Serum PSA and Hemodilution Although examination of equation (5) indicates that the concentration of PSA (ie, y) should be inversely related to V, the magnitude of this hemodilution is not apparent without further details. If we select typical values for α, M, V, ρ, and β, we can use equation (5) to demonstrate how PSA relates to V. For example, let us assume that the weight of the prostate is 53 g, because this is what a study of over 13,000 men found.3 Next, let β assume its mean value of .26 (1/day) found by following PSA levels after prostatectomy,7 and let ρ have the value of .00077, which was the mean found at the time of diagnosis of prostate cancer for 100 men.8 Let V assume its mean value of 3,063 mL.4,5 For a range of PSA values from 1 to 15 ng/mL, the corresponding median value for α would be 121 (1/(ng × day)). Using these values and allowing V to vary, Figure 2 demonstrates how serum PSA (y-axis) can be affected by hemodilution. Furthermore, by fixing height at 168 cm (mean of the data for men in Figure 1), Figure 3 demonstrates how serum PSA can be affected by body weight (x-axis). Although these results flow naturally from the kinetic model, similar results have also been obtained empirically from patient data.3,10 Figure 2 View largeDownload slide Plot of the expected relationship between serum prostate-specific antigen and plasma volume as obtained from the kinetic model leading to equation (5). Figure 2 View largeDownload slide Plot of the expected relationship between serum prostate-specific antigen and plasma volume as obtained from the kinetic model leading to equation (5). Figure 3 View largeDownload slide Plot of the expected relationship between serum prostate-specific antigen and body weight as obtained from the kinetic model leading to equation (5). Figure 3 View largeDownload slide Plot of the expected relationship between serum prostate-specific antigen and body weight as obtained from the kinetic model leading to equation (5). PSA Mass and PSA Mass Density Multiplying both sides of equation (5) by V yields an equation for PSA mass (PSAM) as follows: PSAM = y * V =α * M(ρ + β) (6) Thus, unlike serum PSA, PSAM at least over small time periods reflects just α, W, ρ, and β. In this way, the kinetic model predicts that PSAM should at least partly correct for the effects of hemodilution. Empirical data have verified this result.3,11-14 Next, dividing both sides of equation (5) by M yields an equation for PSA mass density (PSAMD) as follows: PSAMD = y * V / M ~ α / (ρ + β) (7) Equation (7) demonstrates that PSAMD reflects just the quotient α / (ρ + β), so that it brings us one step closer to knowing the value of α, the variable most closely related to the amount and grade of tumor. Note, however, that both numerator and denominator contain variables (α and ρ) that reflect the mass of tumor. If prostate mass is estimated by ultrasound and if plasma volume can be estimated from height, weight, and Hct, then PSAMD can be calculated as PSA * V / M. Thus, PSAMD may become a better prognosticator than PSA. In fact, some have found that PSAMD is more closely related to tumor volume than PSAM.12 Nevertheless, PSAMD remains inversely related to the sum (ρ + β). Thus, higher levels of PSAMD could be due to a higher level of α or to lower levels of ρ and β. Furthermore, because one study demonstrated that removal of PSA takes place in the liver,15 it is possible that varying levels of β could be due to variations in liver function. Discussion The above analysis provides a physiologic background for understanding how the amount of any analyte can be affected by body weight, and the results suggest that obesity-related hemodilution may be at least partly corrected for by multiplying the concentration by an estimate of plasma volume to yield the amount of analyte in units of mass. Undoubtedly, the product introduces noise due to the level of error documented in Figure 1, and it is likely that additional noise comes from unaccounted levels of α1 antichymotrypsin. Nevertheless, some have found that PSAM and PSAMD provide useful prognostic information.11-14 Other laboratory analytes that may be candidates for converting to mass units include hemoglobin, creatinine, glucose, troponins, and even blood cell counts. For example, absolute counts of monocytes can be multiplied by an estimate of blood volume to yield an overall monocyte mass. The early successes for PSAM and PSAMD suggest that more emphasis be given to the importance of plasma volume in the obese population. For example, it is possible that additional studies relating plasma volume to height and weight could improve on the results from Table 2. Finally, the results suggest that laboratorians undertake additional empirical studies to compare how patient outcomes relate to the amount of analyte in units of mass vs units of concentration. References 1. Vollmer RT. Use of serum markers to measure acute myocardial infarct size: lessons of a nonlinear dynamical model. J Theor Biol . 1997; 187: 195- 205. Google Scholar CrossRef Search ADS PubMed 2. Vollmer RT, Humphrey PA. Tumor volume in prostate cancer and serum prostate-specific antigen: analysis from a kinetic viewpoint. Am J Clin Pathol . 2003; 119: 80- 89. Google Scholar CrossRef Search ADS PubMed 3. Bañez LL, Hamilton RJ, Partin AWet al. Obesity-related plasma hemodilution and PSA concentration among men with prostate cancer. JAMA . 2007; 298: 2275- 2280. Google Scholar CrossRef Search ADS PubMed 4. Gibson JG, Evans WA. Clinical studies of the blood volume, II: the relation of plasma and total blood volume to venous pressure, blood velocity rate, physical measurements, age and sex in ninety normal humans. J Clin Invest . 1937; 16: 317- 328. Google Scholar CrossRef Search ADS PubMed 5. Retzlaff JA, Tauxe WN, Kiely JMet al. Erythrocyte volume, plasma volume, and lean body mass in adult men and women. Blood . 1969; 33: 649- 661. Google Scholar PubMed 6. Ogden CL, Carroll MD, Fryar CDet al. Prevalence of Obesity Among Adults and Youth: United States, 2011-2014 . Hyattsville, MD: National Center for Health Statistics; 2015. NCHS Data Brief No. 219. 7. Humphrey PA, Vollmer RT. Relationships between serum prostate-specific antigen and histopathologic appearances of prostate carcinoma. In: Foster CS, Bostwick DG, eds. Pathology of the Prostate . Philadelphia: WB Saunders; 1998: 253- 281. 8. Vollmer RT. Dissecting the dynamics of serum prostate-specific antigen. Am J Clin Pathol . 2010; 133: 187- 193. Google Scholar CrossRef Search ADS PubMed 9. Vollmer RT, Dawson NA, Vogelzang NJ. The dynamics of prostate specific antigen in hormone refractory prostate carcinoma: an analysis of cancer and leukemia group B study 9181 of megestrol acetate. Cancer . 1998; 83: 1989- 1994. Google Scholar CrossRef Search ADS PubMed 10. Baillargeon J, Pollock BH, Kristal ARet al. The association of body mass index and prostate-specific antigen in a population-based study. Cancer . 2005; 103: 1092- 1095. Google Scholar CrossRef Search ADS PubMed 11. Grubb RLIII, Black A, Izmirlian Get al. ; PLCO Project Team. Serum prostate-specific antigen hemodilution among obese men undergoing screening in the prostate, lung, colorectal, and ovarian cancer screening trial. Cancer Epidemiol Biomarkers Prev . 2009; 18: 748- 751. Google Scholar CrossRef Search ADS PubMed 12. Bryniarski P, Paradysz A, Fryczkowski M. PSA mass as a marker of prostate cancer progression after radical prostatectomy. Med Sci Monit . 2011; 17: CR104- CR109. Google Scholar CrossRef Search ADS PubMed 13. Kryvenko ON, Epstein JI, Meier FAet al. Correlation of high body mass index with more advanced localized prostate cancer at radical prostatectomy is not reflected in PSA level and PSA density but is seen in PSA mass. Am J Clin Pathol . 2015; 144: 271- 277. Google Scholar CrossRef Search ADS PubMed 14. Kryvenko ON, Diaz M, Matoso Aet al. Prostate-specific antigen mass density: a measure predicting prostate cancer volume and accounting for overweight and obesity-related prostate-specific antigen hemodilution. Urology . 2016; 90: 141- 147. Google Scholar CrossRef Search ADS PubMed 15. Agha AH, Schechter E, Roy JBet al. Prostate specific antigen is metabolized in the liver. J Urol . 1996; 155: 1332- 1335. Google Scholar CrossRef Search ADS PubMed © American Society for Clinical Pathology, 2018. All rights reserved. For permissions, please e-mail: email@example.com
American Journal of Clinical Pathology – Oxford University Press
Published: Mar 1, 2018
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