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Regular Article |
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
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Abstract |
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Introduction |
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However, the correlation between serum PSA and the volume of cancer is poor and the variance is high in patients with prostate carcinoma. Because of the high variance, several modifications have been proposed to improve the diagnostic accuracy of serum PSA levels. These modifications include the use of age-specific reference ranges, PSA density, PSA velocity, and assays that specifically measure the different molecular forms of PSA.3,5 PSA density is the serum PSA value divided by the ultrasound-determined volume of the prostate.6 PSA velocity is the rate of increase in serum PSA concentrations throughout time.7,8 The molecular forms of PSA that are produced in different ratios in patients with prostate cancer than in those without demonstrable cancer include complexed PSA and free PSA. However, the prospect that these modifications of PSA assays would provide more reliable markers of prostate cancer than a simple serum PSA assay has not yet been convincingly demonstrated.6,8
One approach to understanding the contribution of potential independent variables to predict the serum PSA concentrations in individual patients would involve correlation of the volume of prostate carcinomas with serum PSA values throughout time. This approach would entail measuring the volume of carcinoma in prostates in men in vivo. Unfortunately, the volume of carcinoma in the human prostate cannot be accurately measured in vivo. Imaging techniques such as ultrasound were first thought to be tools that could identify carcinoma in vivo. This prospect has not been realized. The correlation coefficient between ultrasound-determined volume of cancer and pathologically measured volume is as low as 0.1.9 The difficulty of determining the volume of prostate cancer is also reflected in the correlation between location and volume of cancer in needle biopsies and in the radical prostatectomy specimen. Sampling of individual prostates by needle biopsies is too poor to provide an accurate assessment of cancer volume in individual prostates; correlation coefficients as low as 0.5 have been reported.10
We hypothesized that differences in growth rates of prostate cancers could help explain the variance in correlations of serum PSA concentrations with tumor size. Because the volume of human prostate cancer cannot be determined in patients with accuracy, we worked with xenografts of human prostate cancers in immunocompromised mice. We developed a mathematical model that accounted for the contribution of independent variables to the size of the xenograft and the serum level of PSA.
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Methods |
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Development of Xenografts
Details of the xenografts and the method for generating them have been previously published.11 In brief, the LuCaP 23 series of prostate carcinoma xenografts came from a 63-year-old white male donor diagnosed with advanced prostate cancer.2,11 The patient was treated with external beam radiation therapy to the pelvis, chemotherapy, and hormone therapy.2,11 The tumor progressed after hormone ablation therapy. Samples of tumor were removed in a sterile manner from two lymph nodes and a liver metastasis and implanted into SCID mice. Three distinct sublines of these xenografts have been established and serially passaged through at least three generations of SCID mice. These xenografts, which have distinctive functional properties, are termed LuCaP 23.1, 23.8, and 23.12, respectively. Histologically, all three xenografts are adenocarcinomas. They differ with respect to growth rate in both androgen-deprived and androgen-supplemented host mice.
PSA Assays
Serum PSA and xenograft volume data were collected weekly. To measure the concentration of human PSA in the blood of mice with xenografts, 0.2 ml of whole blood were removed from the tail vein by capillary pipette. After centrifuging to separate the serum from the blood cells, the serum sample was analyzed for PSA using an indirect enzyme-linked immunosorbent assay. This assay is specific for human PSA. The blood of mice lacking xenografts of human prostate tissue, either benign or malignant, have no detectable PSA. This assay detects concentrations of PSA as low as 0.05 µg/ml.11
Anatomy of Xenografts
Being located subcutaneously as spherical tumor masses, the size
of xenografts can readily and accurately be measured using calipers
opposed to the skin on either side of the xenograft. Histologically,
the structure and cell composition of different xenografts is virtually
identical, regardless of the size, growth rate, or androgen sensitivity
of the xenograft. The majority (>95%) of cells are prostate
carcinoma. Host mouse inflammatory, stromal, and vascular cells
represent 
5% of cells in the tumors.
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Results |
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Our mathematical model for serum PSA dynamics can be written, in
words, as the conservation equation: the rate of change of PSA equals
the source of PSA from benign cells plus the source of PSA from cancer
cells minus the loss of PSA from the blood.
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 | (1) |
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For our analysis, we assume that the total volume of benign
(noncancerous) PSA-producing cells is constant
(Vh = constant) in men with
prostate carcinoma and that the total tumor cell population increases
exponentially in both patients and in tumor xenografts. Because we will
compare the model results to experimental data received from xenografts
in mice, we assume that an initial tumor volume
V0 is implanted in the mouse and that
at some future time, t, the total tumor volume
Vc is defined by the equation
Vc(t) =
V0
e
t where
is the tumor growth rate such that ln (2)/ is the tumor doubling
time. Because we are considering data from mice, we assume that the
initial serum PSA level, p0, is zero.
Figure 1
shows the range of behavior that
is possible under the proposed mathematical model. Given exponential
tumor growth (dotted line), serum PSA levels (solid lines) can either
increase concurrently with the tumor volume as in Figure 1a
or appear
delayed until significant tumor volume has accrued as in Figure 1b
. The
parameter differentiating these two scenarios is µ, the ratio of the
PSA loss rate, , to the tumor growth rate, . When µ is large,
the serum PSA level increases with tumor volume (Figure 1a)
. Otherwise,
there is an apparent lag between tumor growth and elevated serum PSA
levels (Figure 1b)
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shows serum PSA level predicted
by the model for a tumor of given volume V versus
the tumor growth rate. For any given tumor volume V, the
serum PSA level varies with the growth rate. Two tumors diagnosed at
the same volume (V) can manifest very different serum
PSA levels depending on the length of time it took them to grow to
volume V.
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Extensive analysis of the three different xenografts of LuCaP 23 in laboratory mice by Ellis and colleagues11 provides experimental results for comparison with our model predictions.
Untreated Tumor Growth and Serum PSA Levels
The LuCaP 23 xenograft cell sublines were implanted into 10 male
(nude) mice. Tumor volume and serum PSA data are shown in Figure 3
as previously reported by Ellis and
colleagues. 11
The circles in Figure 3
represent the mean
experimental values at each time point with standard deviations defined
by the bars attached to the data points. Linear regression analysis of
the log of tumor volume versus time revealed tumor growth
rates of 0.0655, 0.0504, and 0.0487 (1/day) for LuCaP 23.1, 23.8,
and 23.12, respectively. The fit of the exponential growth curves to
the experimental data are given in Figure 3
; a, c, and e.
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, and the production rate of PSA per unit volume of
cancerous cells per unit time, ßc. We do not
need to know the production rate of PSA per unit volume of normal human
prostate cells per unit time ßh because there
are no normal human PSA-producing prostate cells in the mice with
xenografts.
Ellis and colleagues11
cite a mean PSA half-life of 12.9
hours for the LuCaP 23 studies described above. We convert this
half-life to a PSA decay rate as = log(2)/12.9
hours = 0.0537/hour = 1.2896/day. To calculate the production
of PSA per unit volume of tumor tissue per unit time,
ßc, we consider Figure 4
suggesting that the ratio of the serum
PSA (p) to the tumor volume
(Vc) is asymptotic to a
constant for large tumor volumes:
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the PSA index
and cite estimates of 1.27, 1.63, and 5.21
ng/ml/mm3
for LuCaP 23.1, 23.8, and 23.12,
respectively. We then obtain estimates for ßc
by the formula
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. Using these estimates, we plot serum
PSA levels predicted by the model in Figure 3, b, d, and f
. Note that
although the serum PSA level seems to increase exponentially with tumor
volume (Vc) for later times,
the solution to Equation 1
for the xenograft case
(Vh=0,p0=0):
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Discussion |
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. The model suggests that this delay depends on the parameter µ
defined as the PSA decay rate normalized by the tumor growth rate
. When µ is small, the tumor growth rate is large compared to the
PSA decay rate and a large tumor could yield a low serum PSA level, at
least for a short time. Alternatively, when µ is large, the serum PSA
level can increase significantly in sequence with tumor growth. Virtually invariably, serum PSA levels will increase to clinically significant values for each tumor, assuming, of course, that the patient survives sufficiently long for the tumor to reach that critical volume. Because the serum PSA level that is considered clinically significant is standardized by convention at 4 ng/ml, but the net proliferation rate of the tumor cells is quite variable, the model suggests that there can be a large difference in the size of tumors when using serum PSA levels as a screening assay.
To understand the distinction between the two cases of µ small or large, consider two tumors of volume V. Tumor 1 grows quickly and has attained its present volume V in 1 unit of time. Alternatively, tumor 2 grows more slowly and has taken 10 units of time to attain volume V. A cell in slowly growing tumor 2 has been producing PSA for at most 10 units of time. This is 10 - 1 = 9 units of time more than any cell in tumor 1. This increased opportunity for a cell in tumor 2 to produce PSA augments the serum PSA level. Although the tumor 1 is the more rapidly growing tumor, tumor 2 has been leaking PSA for a longer period of time. The more slowly growing tumor 2 has an increased opportunity to produce significant levels of PSA.
From the experimental data presented in Figure 3
, we see that PSA
levels are related to tumor growth. We also observed that a wide range
of PSA levels is observed for different cancers. There are, however,
limitations to our modeling approach. We have restricted our discussion
to the experimental data available from subcutaneous xenografts. These
data presumably only approximate tumor growth within the human
prostate. Additionally, the amount of tumor necrosis, cellular growth
rates, and PSA kinetic rates of xenografts could significantly differ
from values for these parameters in patients with primary prostate
carcinomas. Furthermore, in limiting our study to the three available
different sublines of prostatic tumors that were derived from a single
patient, we may not be modeling the full range of clinical
manifestations of prostate cancer that occur in different patients.
Despite the numerous limitations, this model has led to a better understanding of PSA chemistry and observations of temporal changes of serum PSA values in men with prostate carcinoma that do not seem intuitive on first reflection. Clinically, the fact that patients with large, rapidly growing tumors often have low levels of serum PSA has been perplexing. Our model has devised a possible explanation for the experimentally and clinically observed disparity between tumor volume and serum PSA level.
This is a significant result because our model provides validation that
serum PSA does not predict tumor volume, but is dependent to a
significant degree on the growth rate of the tumor. In Figure 2
we see
that the serum PSA level for a tumor with volume V can extend over a
wide range of possible values, depending on the net growth rate of the
tumor. That prostate cancers may differ markedly in their growth rate
has been demonstrated in patients. Although the rate of growth of
primary tumors in patients cannot be measured directly because they
cannot be imaged, as discussed above,9
surrogate measures
of growth show wide variance among primary prostate carcinomas of the
same stage. For example, the percentage of cells in the S phase of the
cell cycle ranges from 0 to 15%.12
The percentage of
cells in the cell cycle, which is a value that is obtained by counting
the fraction of cells immunohistochemically expressing the cell cycle
marker Ki67, ranges from 0 to 33%.13
There is a
comparably wide range of rates of tumor cell death in different tumors.
Between 0.25 and 5.25 cells per day per cc undergo apoptosis in
different primary prostate carcinomas.14
We have developed a mathematical model to describe a possible mechanism for serum PSA levels as a function of tumor volume. In fact, our simple model has suggested why a rapidly growing tumor does not predictably lead to an increased PSA level. We are optimistic that further mathematical modeling will help quantify these results and assist in determining the best measurement of PSA to be used to indicate tumor growth.
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Footnotes |
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Supported by United States National Science Foundation grants BIR-9256532, DMS-9902385, and DMS-9500766 (to K. R. S.); the National Institutes of Health grant 2P41-RR-01243-1177 (to J. D. M.); the Academic Pathology Fund (to K.R.S.), the Richard M. Lucas Foundation and the National Institutes of Diabetes, Digestive, Kidney Diseases O’Brien Center grant (to R. V. and K. R. B.) and the CapCURE Foundation (L.D.T.).
Accepted for publication March 13, 2001.
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References |
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