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A Quantitative Model for the Dynamics of Serum Prostate-Specific Antigen as a Marker for Cancerous Growth

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A Quantitative Model for the Dynamics of SerumProstate-Specific Antigen as a Marker for CancerousGrowth
An Explanation for a Medical Anomaly
Kristin R. Swanson,*
†
Lawrence D. True,
‡
Daniel W. Lin,
§
Kent R. Buhler,
§
Robert Vessella,
§
and James D. Murray
†
From the Department of Pathology,
*
Laboratory of Neuropathology, Harborview Medical Center, Seattle; and the Departments of Applied Mathematics,
†
Pathology,
‡
and Urology,
§
University of Washington, Seattle, Washington
Prostate-specific antigen (PSA) is an enzyme pro-duced by both normal and cancerous prostate epithe-lial cells. Although PSA is the most widely used serum marker to detect and follow patients with prostaticadenocarcinoma, there are certain anomalies in the values of serum levels of PSA that are not understood. We developed a mathematical model for the dynamicsof serum levels of PSA as a function of the tumor volume.Ourmodelresultsshowgoodagreementwith experimental observations and provide an explana-tion for the existence of significant prostatic tumor mass despite a low-serum PSA. This result can be very useful in enhancing the use of serum PSA levels as a marker for cancer growth.
(Am J Pathol 2001, 158:2195–2199)
Prostate-specific antigen (PSA) is a proteolytic enzymethat is used widely as a tumor marker for the diagnosis ofprostate cancer and for monitoring patients with prostaticadenocarcinoma. PSA is produced by normal humanprostate secretory epithelial cells and by virtually all pros-tatic adenocarcinomas.
1–3
The serum concentration ofPSA correlates with the age of the patient, with the size ofthe prostate in men without demonstrable prostate carci-noma, with the volume of carcinoma in men with bothprimary and metastatic prostate carcinoma,
1
and with thestage of prostate cancer.
3,4
The presumed mechanismfor the observation that serum PSA levels in men withprostate carcinoma are significantly higher, on average,than levels in men without carcinoma demonstrable byprostate needle biopsies is that PSA leaks from malignantcells and glands into the interstitium and thence into theblood instead of being confined to the ductal excretorysystem.However, the correlation between serum PSA and thevolume of cancer is poor and the variance is high inpatients with prostate carcinoma. Because of the highvariance, several modifications have been proposed toimprove the diagnostic accuracy of serum PSA levels.These modifications include the use of age-specific ref-erence ranges, PSA density, PSA velocity, and assaysthat specifically measure the different molecular forms ofPSA.
3,5
PSA density is the serum PSA value divided bythe ultrasound-determined volume of the prostate.
6
PSAvelocity is the rate of increase in serum PSA concentra-tions throughout time.
7,8
The molecular forms of PSA thatare produced in different ratios in patients with prostatecancer than in those without demonstrable cancer in-clude complexed PSA and free PSA. However, the pros-pect that these modifications of PSA assays would pro-vide more reliable markers of prostate cancer than asimple serum PSA assay has not yet been convincinglydemonstrated.
6,8
One approach to understanding the contribution ofpotential independent variables to predict the serum PSAconcentrations in individual patients would involve corre-lation of the volume of prostate carcinomas with serumPSA values throughout time. This approach would entailmeasuring the volume of carcinoma in prostates in men
invivo
.Unfortunately,thevolumeofcarcinomainthehumanprostate cannot be accurately measured
in vivo
. Imagingtechniques such as ultrasound were first thought to betools that could identify carcinoma
in vivo
. This prospecthas not been realized. The correlation coefficient be-tween ultrasound-determined volume of cancer and patho-logicallymeasuredvolumeisaslowas0.1.
9
Thedifficultyof
Supported by United States National Science Foundation grants BIR-9256532, DMS-9902385, and DMS-9500766 (to K. R. S.); the NationalInstitutes of Health grant 2P41-RR-01243-1177 (to J. D. M.); the AcademicPathology Fund (to K.R.S.), the Richard M. Lucas Foundation and theNational Institutes of Diabetes, Digestive, Kidney Diseases O’Brien Centergrant (to R. V. and K. R. B.) and the CapCURE Foundation (L.D.T.).Accepted for publication March 13, 2001.Address reprint requests to Kristin R. Swanson, Ph.D., University ofWashington, Harborview Medical Center, Department of Pathology andLaboratory of Neuropathology, 325 9th Ave., Box 35971, Seattle, WA98104. E-mail: swanson@amath.washington.edu.
American Journal of Pathology, Vol. 158, No. 6, June 2001Copyright © American Society for Investigative Pathology
2195
determining the volume of prostate cancer is also reflectedin the correlation between location and volume of cancer inneedlebiopsiesandintheradicalprostatectomyspecimen.Sampling of individual prostates by needle biopsies is toopoor to provide an accurate assessment of cancer volumein individual prostates; correlation coefficients as low as 0.5have been reported.
10
We hypothesized that differences in growth rates ofprostate cancers could help explain the variance in cor-relations of serum PSA concentrations with tumor size.Because the volume of human prostate cancer cannot bedetermined in patients with accuracy, we worked withxenograftsofhumanprostatecancersinimmunocompro-mised mice. We developed a mathematical model thataccounted for the contribution of independent variablesto the size of the xenograft and the serum level of PSA.
Methods
To better understand the relationship between serumPSA and the rate of tumor growth, we have developed amathematical model for serum PSA dynamics that isbased on studies of xenografts of human prostate carci-noma implanted in severe combined immunodeficiency(SCID) mice.
Development of Xenografts
Details of the xenografts and the method for generatingthem have been previously published.
11
In brief, theLuCaP 23 series of prostate carcinoma xenografts camefrom a 63-year-old white male donor diagnosed with ad-vanced prostate cancer.
2,11
The patient was treated withexternal beam radiation therapy to the pelvis, chemother-apy, and hormone therapy.
2,11
The tumor progressedafter hormone ablation therapy. Samples of tumor wereremoved in a sterile manner from two lymph nodes and aliver metastasis and implanted into SCID mice. Threedistinct sublines of these xenografts have been estab-lished and serially passaged through at least three gen-erations of SCID mice. These xenografts, which havedistinctive functional properties, are termed LuCaP 23.1,23.8, and 23.12, respectively. Histologically, all threexenografts are adenocarcinomas. They differ with re-spect to growth rate in both androgen-deprived and an-drogen-supplemented host mice.
PSA Assays
Serum PSA and xenograft volume data were collectedweekly. To measure the concentration of human PSA inthe blood of mice with xenografts, 0.2 ml of whole bloodwere removed from the tail vein by capillary pipette. Aftercentrifuging to separate the serum from the blood cells,the serum sample was analyzed for PSA using an indirectenzyme-linked immunosorbent assay. This assay is spe-cific for human PSA. The blood of mice lacking xeno-grafts of human prostate tissue, either benign or malig-nant, have no detectable PSA. This assay detectsconcentrations of PSA as low as 0.05
g/ml.
11
Anatomy of Xenografts
Being located subcutaneously as spherical tumormasses, the size of xenografts can readily and accuratelybe measured using calipers opposed to the skin on eitherside of the xenograft. Histologically, the structure and cellcomposition of different xenografts is virtually identical,regardless of the size, growth rate, or androgen sensitiv-ity of the xenograft. The majority (
95%) of cells areprostate carcinoma. Host mouse inflammatory, stromal,and vascular cells represent
5% of cells in the tumors.
Results
Mathematical Model
Our mathematical model for serum PSA dynamics can bewritten, in words, as the conservation equation: the rate ofchange of PSA equals the source of PSA from benigncells plus the source of PSA from cancer cells minus theloss of PSA from the blood.Rate of change of PSA
source of PSA from benign cells
source of PSA from cancer cells
loss of PSA from the bloodTo introduce the relevant parameters, we write the modelmathematically asdpdt
h
V
h
c
V
c
p, (1)where
p
(
t
) is the serum PSA level at time
t
,
V
h
and
V
c
arethe volumes of benign and cancerous PSA-producingcells, respectively. Initially, at time
t
0, we assume theserum PSA level is
p
0
. We assume that PSA is producedby benign and cancerous cells at the rate
h
and
c
,respectively. Cancer cells leak much more PSA thanbenign cells so
h
is much smaller than
c
:
h
c
. Weassume that PSA is metabolized and cleared from theblood of both humans and mice with xenografts at ap-proximately the same constant rate
.For our analysis, we assume that the total volume ofbenign (noncancerous) PSA-producing cells is constant(
V
h
constant) in men with prostate carcinoma and thatthe total tumor cell population increases exponentially inboth patients and in tumor xenografts. Because we willcompare the model results to experimental data receivedfrom xenografts in mice, we assume that an initial tumorvolume
V
0
is implanted in the mouse and that at somefuture time,
t
, the total tumor volume
V
c
is defined by theequation
V
c
(t)
V
0
e
t
where
is the tumor growth ratesuch that ln (2)/
is the tumor doubling time. Because weare considering data from mice, we assume that the initialserum PSA level,
p
0
, is zero.Figure 1 shows the range of behavior that is possibleunder the proposed mathematical model. Given expo-
2196 Swanson et al
AJP June 2001, Vol. 158, No. 6
nential tumor growth (dotted line), serum PSA levels (sol-id lines) can either increase concurrently with the tumorvolume as in Figure 1a or appear delayed until significanttumor volume has accrued as in Figure 1b. The param-eter differentiating these two scenarios is
, the ratio ofthe PSA loss rate,
, to the tumor growth rate,
. When
is large, the serum PSA level increases with tumor volume(Figure 1a). Otherwise, there is an apparent lag betweentumor growth and elevated serum PSA levels (Figure 1b).Figure 2 shows serum PSA level predicted by themodel for a tumor of given volume
V versus
the tumorgrowth rate. For any given tumor volume
V
, the serumPSA level varies with the growth rate. Two tumors diag-nosed at the same volume (
V
) can manifest very differentserum PSA levels depending on the length of time it tookthem to grow to volume
V
.
Comparison with Experimental Data: LuCap 23
Extensive analysis of the three different xenografts ofLuCaP 23 in laboratory mice by Ellis and colleagues
11
provides experimental results for comparison with ourmodel predictions.
Figure 3.
Experimental observations of LuCaP 23 tumor volume (mm
3
) andserum PSA levels (ng PSA per ml of serum) in xenografts with modelpredictions
versus
time (in days).
a
,
c
, and
e:
Tumor volumes for LuCaP 23.1,23.8, 23.12, respectively.
b
,
d
, and
f:
Serum PSA levels for LuCaP 23.1, 23.8,23.12, respectively.
Figure 1.
Range of behavior of PSA production given exponential tumor growth. The PSA level (
solid line
) can either increase significantly in sequence withtumor growth (
a
,
dotted line
) or appear delayed depending on the parameter
defined as the ratio of the loss rate of PSA,
, and the growth rate of the cancermass,
(
b
).
Figure 2.
Serum PSA level, p, for a fixed tumor volume
V versus
the tumorgrowth rate
. For a given tumor volume, a wide range of serum PSA levelscan be observed depending on the length of time it took the tumor to grow to volume V.
Quantitative Model for PSA Dynamics 2197
AJP June 2001, Vol. 158, No. 6
Untreated Tumor Growth and Serum PSALevels
The LuCaP 23 xenograft cell sublines were implanted into10 male (nude) mice. Tumor volume and serum PSA dataare shown in Figure 3 as previously reported by Ellis andcolleagues.
11
The circles in Figure 3 represent the meanexperimental values at each time point with standarddeviations defined by the bars attached to the datapoints. Linear regression analysis of the log of tumorvolume
versus
time revealed tumor growth rates
of0.0655, 0.0504, and 0.0487 (1/day) for LuCaP 23.1, 23.8,and 23.12, respectively. The fit of the exponential growthcurves to the experimental data are given in Figure 3; a,c, and e.To predict the serum PSA levels as a function of timegiven the tumor growth characteristics of a specific sub-line, we need estimates of the PSA loss rate,
, and theproduction rate of PSA per unit volume of cancerous cellsper unit time,
c
. We do not need to know the productionrate of PSA per unit volume of normal human prostatecells per unit time
h
because there are no normal humanPSA-producing prostate cells in the mice with xenografts.Ellis and colleagues
11
cite a mean PSA half-life of 12.9hours for the LuCaP 23 studies described above. Weconvert this half-life to a PSA decay rate as
log(2)/ 12.9 hours
0.0537/hour
1.2896/day. To calculate theproduction of PSA per unit volume of tumor tissue per unittime,
c
, we consider Figure 4 suggesting that the ratio ofthe serum PSA (
p
) to the tumor volume (
V
c
) is asymptoticto a constant for large tumor volumes:
pV
c
c
.Ellis and colleagues
11
term this ratio of the serum PSAlevel,
p
, and the tumor growth rate
the PSA index andcite estimates of 1.27, 1.63, and 5.21 ng/ml/mm
3
forLuCaP 23.1, 23.8, and 23.12, respectively. We then ob-tain estimates for
c
by the formula
c
PSA indexWe tabulate the LuCaP 23 series parameter estimates inTable 1. Using these estimates, we plot serum PSA levelspredicted by the model in Figure 3, b, d, and f. Note thatalthough the serum PSA level seems to increase expo-nentially with tumor volume (
V
c
) for later times, the solu-tion to Equation 1 for the xenograft case (
V
h
0,
p
0
0):
p
(
t
)
c
V
c
e
t
c
V
0
e
t
e
t
has a biexponential form that manifests a small bumpnear
t
0 in the plots of the serum PSA levels in Figure3, b, d, and f.
Discussion
The mathematical model suggests that there could be anapparent delay between tumor growth and PSA produc-tion as illustrated by Figure 1. The model suggests thatthis delay depends on the parameter
defined as thePSA decay rate
normalized by the tumor growth rate
.When
is small, the tumor growth rate is large comparedto the PSA decay rate and a large tumor could yield a lowserum PSA level, at least for a short time. Alternatively,when
is large, the serum PSA level can increase sig-nificantly in sequence with tumor growth.Virtually invariably, serum PSA levels will increase toclinically significant values for each tumor, assuming, ofcourse, that the patient survives sufficiently long for thetumor to reach that critical volume. Because the serumPSA level that is considered clinically significant is stan-dardized by convention at 4 ng/ml, but the net prolifera-tion rate of the tumor cells is quite variable, the modelsuggests that there can be a large difference in the sizeof tumors when using serum PSA levels as a screeningassay.
Figure 4.
PSA index, the ratio of the serum PSA level,
p
, to the tumor volume,
V
c
,
versus
the tumor volume,
V
c
.
Table 1.
Parameter Estimates for the LuCaP 23 Experiments
Parameter SymbolLuCaP23.1LuCaP23.8LuCaP23.12 UnitsTumor growth rate
0.0655 0.0504 0.0487 1/dayNormal PSA production
h
0 0 0 ng/mm
3
/dayCancer PSA production
c
1.7210 2.1841 6.9722 ng/mm
3
/dayDecay rate of PSA (loss)
1.2896 1.28966 1.2896 1/dayImplanted tumor volume
V
0
20–25 20–25 20–25 mm
3
Volume of benign PSA-producing cells
V
h
0 0 0 mm
3
2198 Swanson et al
AJP June 2001, Vol. 158, No. 6
Tounderstandthedistinctionbetweenthetwocasesof
small or large, consider two tumors of volume
V
. Tumor1 grows quickly and has attained its present volume
V
in1 unit of time. Alternatively, tumor 2 grows more slowlyand has taken 10 units of time to attain volume
V
. A cell inslowly growing tumor 2 has been producing PSA for atmost 10 units of time. This is 10
1
9 units of time morethan any cell in tumor 1. This increased opportunity for acell in tumor 2 to produce PSA augments the serum PSAlevel. Although the tumor 1 is the more rapidly growingtumor, tumor 2 has been leaking PSA for a longer periodof time. The more slowly growing tumor 2 has an in-creased opportunity to produce significant levels of PSA.From the experimental data presented in Figure 3, wesee that PSA levels are related to tumor growth. We alsoobserved that a wide range of PSA levels is observed fordifferent cancers. There are, however, limitations to ourmodeling approach. We have restricted our discussion totheexperimentaldataavailablefromsubcutaneousxeno-grafts. These data presumably only approximate tumorgrowth within the human prostate. Additionally, theamount of tumor necrosis, cellular growth rates, and PSAkinetic rates of xenografts could significantly differ fromvalues for these parameters in patients with primary pros-tate carcinomas. Furthermore, in limiting our study to thethree available different sublines of prostatic tumors thatwere derived from a single patient, we may not be mod-eling the full range of clinical manifestations of prostatecancer that occur in different patients.Despite the numerous limitations, this model has led toa better understanding of PSA chemistry and observa-tions of temporal changes of serum PSA values in menwith prostate carcinoma that do not seem intuitive on firstreflection. Clinically, the fact that patients with large, rap-idly growing tumors often have low levels of serum PSAhas been perplexing. Our model has devised a possibleexplanationfortheexperimentallyandclinicallyobserveddisparity between tumor volume and serum PSA level.This is a significant result because our model providesvalidation that serum PSA does not predict tumor volume,but is dependent to a significant degree on the growthrate of the tumor. In Figure 2 we see that the serum PSAlevel for a tumor with volume V can extend over a widerange of possible values, depending on the net growthrate of the tumor. That prostate cancers may differ mark-edly in their growth rate has been demonstrated in pa-tients. Although the rate of growth of primary tumors inpatients cannot be measured directly because they can-not be imaged, as discussed above,
9
surrogate mea-sures of growth show wide variance among primary pros-tate carcinomas of the same stage. For example, thepercentageofcellsintheSphaseofthecellcyclerangesfrom 0 to 15%.
12
The percentage of cells in the cell cycle,which is a value that is obtained by counting the fractionof cells immunohistochemically expressing the cell cyclemarker Ki67, ranges from 0 to 33%.
13
There is a compa-rably wide range of rates of tumor cell death in differenttumors. Between 0.25 and 5.25 cells per day per ccundergo apoptosis in different primary prostate carcino-mas.
14
We have developed a mathematical model to describea possible mechanism for serum PSA levels as a functionof tumor volume. In fact, our simple model has suggestedwhy a rapidly growing tumor does not predictably lead toan increased PSA level. We are optimistic that furthermathematical modeling will help quantify these resultsand assist in determining the best measurement of PSAto be used to indicate tumor growth.
References
1. Oesterling JE: Prostate specific antigen: a critical assessment of themost useful tumor marker for adenocarcinoma of the prostate. J Urol1991, 145:907–9232. Swanson KR: Mathematical Modeling of the Growth and Control ofTumors, Ph.D. Thesis, University of Washington, 19993. Sokoll L, Chan DW: Total, free, and complexed PSA: analysis andclinical utility. J Clin Ligand Assay 1998, 21:171–1794. Partin AW, Kattan MW, Subong EN, Walsh PC, Wojno KJ, OesterlingJE, Scardino PT, Pearson JD: Combination of prostate-specific anti-gen, clinical stage, and Gleason score to predict pathological stageof localized prostate cancer: a multi-institutional update. JAMA 1997,277:1445–14515. Chan DW: Prostate-specific antigen: advances and challenges. ClinChem 1999, 45:755–7566. Lin DW, Gold MH, Ransom S, Ellis WJ, Brawer MK: Transition zoneprostate specific antigen: lack of use in prediction of prostate carci-noma. J Urol 1998, 160:77–817. Smith DS, Catalona WJ: Rate of change in serum prostate specificantigen levels as a method for prostate cancer detection. J Urol 1994,152:1163–11678. Carter HB, Pearson JD, Waclawiw Z, Metter EJ, Chan DW, Guess HA,Walsh PC: Prostate-specific antigen variability in men without prostatecancer: effect of sampling interval on prostate-specific antigen veloc-ity. Urology 1995, 45:5919. Terris MK, Haney DJ, Johnstone IM, McNeal JE, Stamey TA: Predic-tion of prostate cancer volume using prostate-specific antigen levels,transrectal ultrasound, and systematic sextact biopsies. Urology1995, 45:75–8010. Cupp MR, Bostwick DG, Myers RP, Oesterling JE: The volume ofprostate cancer in biopsy specimen cannot reliably predict the quan-tity of cancer in the radical prostatectomy specimens on an individualbasis. J Urol 1995, 153:1543–154811. Ellis WJ, Vessella RL, Buhler KR, Baldou F, True LD, Bigler SA, CurtisD, Lange PH: Characterization of a novel androgen-sensitive, pros-tate-specific antigen-producing prostatic carcinoma xenograft: luCaP23. Clin Cancer Res 1996, 2:1039–104812. Pollack A, Troncosco P, Zagars GK, von Eschebach AC, Mak AC, WuCS, Terry NH: The significance of DNA-ploidy and s-phase fraction innode-positive (stage D1) prostate cancer treated with androgen ab-lation. Prostate 1997, 31:21–2813. Bubendorf L, Sauter G, Moch H, Schmid HP, Gaser TC, Jordan P,Mihatsch MJ: Ki67 labelling index: an independent predictor of pro-gression in prostate cancer treatment by radical prostatectomy.J Pathol 1996, 178:437–44114. Berges RR, Vukanovic J, Epstein JI, Carmichel M, Cisek K, JohnsonDE, Veltri RW, Walsh PC, Isaacs JT: Implication of cell kineticchanges during the progression of human prostatic cancer. ClinCancer Res 1995, 1:473–480
Quantitative Model for PSA Dynamics 2199
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