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Biomedical Informatics, by Jules J. Berman, cover Perl Programming for Medicine and Biology, by Jules J. Berman, cover Ruby for Medicine and Biology, by Jules J. Berman, cover Ruby: The Programming Language, by Jules J. Berman, cover


















Berman JJ, Moore GW.
Spontaneous regression of residual tumour burden: prediction by Monte Carlo simulation.
Anal Cell Pathol. 1992 Sep;4(5):359-368.

ABSTRACT.

Spontaneous regression of early (preneoplastic) lesions is one of the fundamental mysteries in carcinogenesis. We have hypothesized that lesions regress for strictly probabilistic reasons. Small populations composed of dividing cells that have a non-zero probability of death in any division cycle may either grow or shrink as chance dictates. Complete regression occurs when a small population shrinks to zero cells. In a computerized Monte Carlo simulation, the cell growth of groups of 500 cells with constant generation times and preset prob-abilities of cell death was determined. Regression of proliferating clones (extinction of clones before the 100th generation) was the usual outcome. When the probability of cell death in each cell at each generation is set at 50%, 49%, 48%, 47%, 46%, and 45%, regression of clones before the 100th generation occurs in 98%, 96%, 91%, 86%, 78%, and 75% of clones, respectively. Simulations with probabilities of cell death greater than 50% lead to early clonal extinction. When the probability of cell death is appreciably lower than 45%, clones grow so rapidly that their simulation is not consistent with observed tumor growth. Results suggest that regression of early lesions and initiated cells may occur purely as a probabilistic consequence of intrinsic growth conditions in small populations (early lesions). Cell death probability near 45% produces simulated clonal growth rates similar to empirically observed tumor growth.

Key Words: spontaneous regression, preneoplastic lesion, liver, Monte Carlo simulation, cell death, initiation, promotion.

LONG ABSTRACT.

Spontaneous regression of preneoplastic lesions is a fundamental mystery of carcinogenesis. The Solt-Farber model of liver carcinogenesis is a widely used system for the study of cell population events that lead to cancer. In this system, initiation occurs when rats are exposed to a carcinogen, and promotion of initiated cells is achieved by partial hepatectomy. In the ensuing months, hundreds of preneoplastic lesions arise. Most early lesions regress for reasons that are entirely unknown. The observation of emerging early proliferative lesions followed by regression of most early lesions is also observed in rodent models of skin tumorigenesis. In humans, the regression of dysplastic lesions of the cervix is an accepted clinical phenomenon.

We have hypothesized that lesions regress for strictly probabilistic reasons. Small populations composed of dividing cells that have a non-zero probability of death in any division cycle may either grow or shrink as chance dictates. In a computerized Monte Carlo model of clonal population growth, groups of 500 cells with constant generation times and preset probabilities of cell death was simulated. Regression of proliferating clones (extinction of clones before the 100th generation) was the usual outcome. When the probability of cell death in each cell at each generation is set at 50%, 49%, 48%, 47%, 46%, and 45%, regression of clones before the 100th generation occurs in 98%, 96%, 91%, 86%, 78%, and 75% of clones. Simulations with probabilities of cell death greater than 50% lead to early clonal extinction. When the probability of cell death is appreciably lower than 45%, clones grow so rapidly that their simulation is inconsistent with clinically and experimentally observed rates of tumor growth. Results suggest that regression of preneoplastic lesions and initiated cells may occur purely as a probabilistic consequence of intrinsic growth conditions in small populations (early lesions).

RSUM.

La r‚gression spontan‚e des l‚sions pr‚n‚oplastiques est un mystŠre fondamental de carcinogenŠse. Le modŠle de Solt-Farber de carcinogenŠse du foie est un systŠme, qui est employ‚ de loin pour l'‚tude des ‚v‚nements de la population des cellules qui introduitent le cancer. En ce systŠme initiation a lieu quand les rats sont expos‚s … un carcinogen, et la promotion des cellules initi‚es est r‚alis‚e par une h‚patectomie partielle. Dans les mois suivants, les centains des l‚sions pr‚n‚oplastiques viennent. La plupart des l‚sions pr‚n‚oplastiques regresse … cause des raisons inconnues. L'observation des l‚sions pr‚n‚oplastiques prolif‚ratives ‚mergentes qui suivant par r‚gression de la plupart des l‚sions pr‚n‚oplastiques est vu ainsi dans les modŠles de tumorogenŠse de la peau des rongeurs. Dans les ˆtres humains, la r‚gression des l‚sions dysplastiques du cervix est un ph‚nomŠne clinique accept‚.

Nous avons hypothesis‚ que les l‚sions regressent pour les raisons seulement probabilistiques. Les petites populations des cellules qui divisent et qui ont une probabilit‚ de non-zero … mourir dans un cycle de division peuvent croŒtre ou retr‚cir par hazard. Dans le modŠle computeris‚ de Monte Carlo de la croissance de population, les groupes de cinq cents cellules avec les p‚riodes de g‚n‚rations constantes et les probabilit‚s de la mort des cellules qui sont fix‚es en avance, ‚tait simul‚. La r‚gression des clones qui prolif‚re (l'extinction des clones avant la centiŠme g‚n‚ration) ‚tait de r‚sultat normal. Quand la probabilit‚ de la mort de cellule dans chaque cellule en chaque g‚n‚ration est fix‚e … 50%, 49%, 48%, 47%, 46%, et 45%, la r‚gression des clones avant la centiŠme g‚n‚ration a lieu dans 98%, 96%, 91%, 86%, 78%, et 75% des clones. Les simulations des probabilit‚s de la mort des cellules plus de 50% a fait d'extinction commen‡ante des clones. Quand la probabilit‚ de la morte des cellules est beaucoup moins de 45%, les clones croissent si vite qui leure simulation est inconsistent o— on a remarqu‚ ou cliniquement ou exp‚rimentalement les taux de croissance du tumeur. Les r‚sultats suggŠrent que la r‚gression des l‚sions pr‚n‚oplastiques et des cellules initi‚es puisse avoir lieu seulement comme une cons‚quence probabilistique des conditions de la croissance intrinsique dans les petites populations (l‚sions pr‚n‚oplastiques).

Mots-clefs: r‚gression spontan‚e, l‚sions pr‚n‚oplastiques, foie, simulation de Monte Carlo, mort de la cellule, initiation, promotion.

RESUMEN.

La regresi¢n espont nea de lesiones preneopl sicas es un misterio fundamental de carcinog‚nesis. El modelo Solt-Farber de carcinog‚nesis del higado es un sistema, de amplia aceptaci¢n, usado para el estudio de eventos que generan cancer en poblaciones celulares. En este sistema, la iniciaci¢n ocurre cuando se exponen ratas a un carcin¢geno, y la promoci¢n de c‚lulas iniciadas se lleva a cabo por hepatectom¡a parcial. En los meses siguientes, surgen cientos de lesiones preneopl sicas. La mayor¡a de las lesiones preneopl sicas tienen una regresi¢n por razones enteramenta desconocidas. La observaci¢n de lesiones proliferativas tempranas seguidas par la regresi¢n de la mayor¡a de lesiones preneopl sicas es observada tambi‚n en modelos de tumorog‚nesis en piel en roedores. En humanos, la regresi¢n de lesiones displ sicas del c‚rvix es un fen¢meno clinico aceptado.

Nuestra hip¢tesis es que las lesiones tienen una regresi¢n por razones estrictamente de probabilidad. Peque¤as poblaciones compuestas por c‚lulas en divisi¢n que tienen una probabilidad nocero de muerte en cualquier ciclo de divisi¢n pueden crecer o disminuir de azarosamente. En el modelo computerizado Monte Carlo de poblaci¢n de crecimiento clonal, grupos de 500 c‚lulas con tiempos de generaci¢n constantes y probabilidades de muerte c‚lular fijadas previamente fueron simulados. La regresi¢n de los clones proliferativos (extinci¢n de los clones antes de la generaci¢n 100) fue el resultado usual. Cuando la probabilidad de muerte celular en cada c‚lula a cada generaci¢n es fijada a 50%, 49%, 48%, 47%, 46%, y 45%, la regresi¢n de los clones antes de la generaci¢n 100 ocurre en 98%, 96%, 91%, 86%, 78%, y 75% de los clones. Simulaciones con probabilidades de muerte celular mayores de 50% llevan a la extinci¢n temprana de los clones. Cuando la probabilidad de muerte celular es apreciablemente menor de 45%, los clones crecen tan r pido que su simulaci¢n es inconsistente con los indices de crecimiento tumoral clinicalmente y experimental-mente observados. Los resultados sugieren que la regresi¢n de las lesiones preneopl sicas y de c‚lulas iniciadas puede ocurrir solamente como una consequencia de probabilidad de las condiciones intr¡nsicas de crecimiento en poblaciones peque¤as (lesiones preneopl sicas).

Palabras-claves: regresi¢n espont nea, lesiones preneopl sicas, higado, muerte celular, iniciaci¢n, promoci¢n.

INTRODUCTION.

The Solt-Farber model of liver carcinogenesis is a widely used system for the study of cell population events that lead to cancer [ 1, 2, 3]. In this system, initiation occurs when rats are exposed to a carcinogen (diethylnitrosamine). Promotion of initiated cells is achieved by partially hepatectomizing the rats. In the ensuing months, hundreds of preneoplastic lesions arise. Most early lesions regress. The observation of emerging early proliferative lesions followed by regression of most early lesions is also observed in rodent models of skin tumorigenesis [4, 5]. In humans, the occurrence of dysplastic lesions that may regress without resulting in carcinoma is commonplace (e.g., low-grade cervical dysplasias).

The regression of precancerous lesions is an unsolved and fundamental mystery of carcinogenesis. We hypothesize that early lesion regression is governed by the same principles that limit the growth of established tumors: the intrinsic tumor cell death probability. In experimental studies of tumors, it has been shown that tumor cells are continuously dying, often at a rate that matches their cell growth [6]. Cell death in tumors has been attributed to tumor cells that have outstripped their blood supply. However, it can be inferred that, even in the first few months of growth, a tumor must have a high rate of tumor cell death. For instance, if a single hepatoma cell (30,000 µ3 [7]), with a generation time of 1 day is permitted to double with a zero death probability per generation, the ensuing tumor mass would occupy 1 cubic meter in 45 days. In 55 days, it would grow to be over 1,000 cubic meters. In fact, tumors grow slowly, often over many years, and although tumors and preneoplastic lesions have high proliferative indices, this growth is counter-balanced by cell death.

In this study we employ the Monte Carlo method to simulate the growth of cell clones with varied probabilities of cell death [8, 9]. The Monte Carlo method obtains a distribution of outcomes for a mathematical experiment, using a computerized pseudorandom number generator as a substitute for the probability value in the theoretical distribution. Monte Carlo simulations are particularly useful in predicting outcomes where a simple set of initial conditions results in a large number of possible outcomes, from which it is too difficult to obtain analytic solutions. By performing repeated trials and observing how outcomes evolve over time, one can make predictions for complex systems that defy direct evaluation. In this report, the simulated growth of 500 initiated cells was computed using one day as the generation time. The one day generation time was chosen because hepatocellular cell lines and regenerating liver cells divide about every twenty-four hours. Varying probabilities of cell death were assigned to each founder cell. Cell death probabilities were varied between 45% and 53% at 1% intervals. For comparison, a deterministic evaluation of cell death was obtained for the same range of cell death probabilities.

MATERIALS AND METHODS.

Cell proliferation models were programmed on an IBM PC/AT compatible computer (COMTEX, 30368 microprocessor, 25MHz, 330 Mb Priam hard disk), using American National Standard MUMPS (MGlobal, Inc., Houston, TX), and the public-domain File Manager (FileMan) database management system of the United States Department of Veterans Affairs.

In a Monte Carlo simulation, a founder clone consisting of a single cell in generation zero divides once to yield two daughter cells. For each daughter cell at generation one, the simulation program independently draws a pseudorandom number between 1 and 100 in a uniform (i.e., equiprobable) distribution. If the pseudorandom number exceeds the death probability, then that cell continues for at least one more cell division; otherwise the cell divides no further. For example, in a clone with death probability 0.40, the cell divides, and each of the two daughter cells is assigned an independent pseudorandom number between 1 and 100, inclusive. If a daughter cell is assigned a pseudorandom number at most 40, then it dies without dividing. If a daughter cell is assigned a pseudorandom number at least 41, then it is capable of at least one additional mitosis. The process continues until no daughter cells can divide further (extinction), or until the experiment is arbitrarily terminated (in this report, at the 100th generation).

In a deterministic model, fractional cells are permitted at each generation, so that no clone dies unless it has a death probability of one. The number of cells at generation k+1, namely c at k+1, equals 2ck × (1-death probability).

A sample output for a single cell with death probability 0.49 per generation is shown in Table 1. There are 62 generations prior to extinction, in which the clone reaches a maximum size of 111 cells in generation 12, and then becomes smaller. Generation zero has one cell, which duplicates to form two cells; both cells obtain pseudorandom numbers greater than 49, allowing them to proceed to the next generation. Generation one has two cells, which duplicate to form four cells; all four cells obtain pseudorandom numbers greater than 49, allowing them to proceed to the next generation. Generation two has four cells, which duplicate to form eight cells; this time, only four of these cells obtain pseudorandom numbers greater than 49, allowing them to proceed to the next generation. Generation 62 has two cells, which duplicate to form four cells. However, none of the four cells obtain pseudorandom numbers greater than 49, and the clone becomes extinct. In a comparable deterministic experiment, there are only 3.41 cells after 62 generations, but the clone never becomes extinct.

RESULTS.

The results of the Monte Carlo simulations for groups of 500 clones are shown in Table 2. When 500 clones of initiated cells, all having a 24 hour generation time and a 50% probability of any cell dying in any cell cycle, half of the cells died within the first generation cycle. The remaining 250 cells gave rise to lesions of varying sizes, all but ten of which terminated before the 100th cell generation (slightly over 3 months of growth). Among the ten clones that survived to the 100th generation, the largest clone reached a size of only 149 cells. When the probability of cell death was decreased, many more clones reached their hundredth generation and clone sizes grew. In all cases, even in the presence of large surviving clones, the extinction of clones was the rule and clonal survival was the exception. Even when the cell death rate was 45%, there were only 127 (out of 500 initial clones) that reached their 100th generation. The clones that survived to the 100th generation had an average size of 24,851 cells. The largest clone was 89,851. A smaller number of simulations was run for a death probability of 44%, because the numbers of surviving cells in clones became so large that the computer time needed to predict the growth outcome of every cell was excessive.

In the deterministic model, in which fractional cells were permitted and thus no cells could become extinct, the total number of surviving cells usually exceeded the surviving cell numbers predicted by Monte Carlo simulation, but outcomes were all at a similar order of magnitude.

DISCUSSION.

In tumor cell growth simulations where cells grow probabilistically, the number of possible outcomes (distributions of descendants) increases exponentially with each generation. Consequently, it becomes impossible to determine all possible outcomes that might arise when the starting cell populations grow. A practical way to achieve some understanding of how complex populations grow is to run growth simulations and observe the patterns that emerge.

Cell death rates (probability of cell death for each cell in each generation) greater than 50% result in few or no clones reaching 100 generations, and clones that grow to 100 generations are small. For simulations run with cell death probabilities between 44% and 50%, small decrements in death probability resulted in large differences in clonal growth. For instance, at the 100th generation of clones growing with a 45% probability of cell death, the surviving clones had an average size of 24,851. If the cell death rate is 1% higher (46%), the average size of the surviving clones drops to 5,734 cells (more than a 4-fold difference). If the cell death rate is 1% lower (44%), the average size of the surviving clones rises to 131,447.

For the simulation at a 44% probability of cell death (per cell per generation), only 30% of the initial clones survived to 100 generations. In other words, 70% of the population regressed. Regression resulted from initial growth conditions without the intervention of any subsequent event to precipitate the regression phenomenon. The maximum-size clone at the 100th generation of a clone with a 44% death probability was 613,500. Using 30,000 microns3 as the size of a hepatocyte, a mass of 613,500 cells would have a volume of approximately 20 mm3. The actual size of a lesion composed of 613,550 cells would be larger, due to allowances for intercellular space and stroma. However, the predicted clonal cell mass at 100 generations (100 days when the generation time is 24 hours) is similar to the observed size of cells at three months in the Solt-Farber model[1].

It is a popular conception that tumors consist of two populations; a proliferating and a nonproliferating population. The non-proliferating population is presumed to consist of differentiated cells that can no longer divide or of dead cells. The proliferating population is regarded as the subset of cells responsible for tumor growth. Strictly speaking, this model is correct, in that any given tumor cell is either capable of dividing or not. However, the assumption that there is a specific proliferative subpopulation in a tumor is untenable. If there were a population that proliferated and gave rise to other proliferating cells, that popuation would soon overtake the tumor (all other cells belonging to the non-proliferating or dead cell population). Subsequently, the tumor would grow with enormous speed. We propose that tumors are composed of a non-proliferating population of cells that have died or that are post-mitotic (differentiated), as well as a population of cells that can divide. However, the cells that can divide may also die, and the probability that they die is a factor influencing tumor growth rate. If the probability of cell death in the proliferating fraction is stable, then the growth fraction of the tumor would remain constant, as the growth rate is determined experimentally by dividing the [3H-thymidine]-labeling index of the whole tumor cell population by the labeling index of the proliferating subpopulation. The labeling index of the proliferating cell population would be proportionately reduced by the death probability for the population of dividing cells.

A one day generation time was chosen because hepatocellular cell lines and regenerating liver cells divide about every twenty-four hours. This generation period does not have to be exact or even close to the actual tumor cell generation time, as clonal extinction outcomes are independent of the tumor cell growth rate. Cell generation times do have bearing on the size of surviving clones, however, but we consider predictions of tumor growth to be generally inaccurate, serving only to illustrate possible (not necessarily likely) model outcomes.

Simulations were performed with cell death probabilities near 50%. In normal tissue that is not undergoing net growth (e.g., epidermis), a 50% probability of cell death (via end-stage differentiation) is the rule. When a basal cell of skin divides, it produces a differentiated skin cell (to replace the sloughed cell of the stratum corneum) and another basal cell (to replace itself). The death probability for a skin basal cell presumably decreases in wound repair. The assumption that tumor cells proliferate with a stable probability of cell death is inferred from the observed properties of early (preneoplastic) lesion growth, tumor growth, and normal tissue growth.

Monte Carlo simulations show that minor decrements in the rate of cell death can produce enormous changes in clonal growth. It is possible that a feature of the malignant phenotype is the acquisition of a less-than-50% cell death probability (for tumor cells). Minor additional perturbations of the tumor cell death probability may account for dramatic increases in the rate of growth of tumors. These Monte Carlo simulations show that early lesion regression occurs when there is clonal growth with an intrinsic probability of cell death per generation.

ACKNOWLEDGMENTS.

We wish to thank Mme. Barbara L. Moore for the French translation and Dr. Manuel Rivero for the Spanish translation of the abstract.

REFERENCES.

1. Solt DB, Cayama E, Tsuda H, Enomoto K, Lee G, Farber E.
Promotion of liver cancer development by brief exposure to dietary 2-acetylaminofluorene plus partial hepatectomy or carbon tetrachloride.
Cancer Res. 1983 Jan;43(1):188-191.
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Solt DB, Cayama E, Sarma DS, Farber E.
Persistence of resistant putative preneoplastic hepatocytes induced by N-nitrosodiethylamine or N-methyl-N-nitrosourea.
Cancer Res. 1980 Apr;40(4):1112-1118.
PMID: 7357542.
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Ogawa K, Solt DB, Farber E.
Phenotypic diversity as an early property of putative preneoplastic hepatocyte populations in liver carcinogenesis.
Cancer Res. 1980 Mar;40(3):725-733.
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Farber E, Solt D, Cameron R, Laishes B, Ogawa K, Medline A.
Newer insights into the pathogenesis of liver cancer.
Am J Pathol. 1977 Nov;89(2):477-482.
PMID: 920780.
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Solt DB, Medline A, Farber E.
Rapid emergence of carcinogen-induced hyperplastic lesions in a new model for the sequential analysis of liver carcinogenesis.
Am J Pathol. 1977 Sep;88(3):595-618.
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Solt DB, Hay JB, Farber E.
Comparison of the blood supply to diethylnitrosamine-induced hyperplastic nodules and hepatomas and to the surrounding liver.
Cancer Res. 1977 Jun;37(6):1686-1691.
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2. Farber E.
Putative precursor lesions: summary and some analytical considerations.
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3. Goldfarb S, Pugh TD.
Multistage rodent hepatocarcinogenesis.
In: Popper H, Schaffner F, eds. Progress in Liver Diseases. Orlando, FL: Grune and Stratton, Inc. 1986;8:597-620.

4. Foulds L.
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5. Foulds L.
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6. Laird AK.
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7. Elias H, Sherrick JC.
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8. Cashwell ED, Everett CJ.
A Practical manual on the Monte Carlo Method for Random Walk Problems.
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9. Berman JJ, Moore GW.
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10. Steel GG.
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TABLE 1.
 INITIAL FOUNDER CLONE:     1 CELL
           DEATH RATE:    49%
      1     2     4     4     6    12    22    32    41    80
     95    95   111    64    65    52    47    45    43    53
     56    65    71    55    63    60    63    56    61    59
     56    57    57    51    48    45    42    49    53    60
     80    81    78    87    84    85    65    64    41    26
     29    25    23    25    22    15    14    12     8     5
      4     4     2
 EXTINCTION


TABLE 1. Sample growth simulation of a single cell with death probability 0.49, starting in generation 0 with a single cell. There is a total of 62 generations prior to extinction. The clone reaches a maximum size of 111 cells in generation 12. The clone size drops to 52 in generation 15, then rises to a second maximum of 87 in generation 43. Generation 62 has two cells, which duplicate to form four cells, but none of the four cells obtain pseudorandom numbers greater than 49, and the clone becomes extinct.

TABLE 2.
PROB. OF       CLONES         AVERAGE         LARGEST CLONE
CELL DEATH     REACHING       SIZE OF         TO REACH 100
(PER CELL      100 GEN-       SURVIVING       GENERATIONS,
 PER CYCLE)    ERATIONS       CLONES,         # OF CELLS
                              # OF CELLS
   0.53           0                -                     -
   0.52           1               10                    10
   0.51           8           19 (S.D. 16)              44
   0.50          10           42 (S.D. 47)             149
   O.49          22         104 (S.D. 121)             488
   0.48          43         312 (S.D. 411)           2,005
   0.47          72         2,336 (S.D. 8,412)      71,245
   0.46         109         5,734 (S.D. 7167)       37,948
   0.45         127         24,850 (S.D. 24,169     99,277
   0.44           *         131,447(S.D. 138,125)  613,510
   0.00         500**       1.3 x 10 to the 30


TABLE 2. Fate of 500 clones of cells at generation 100, all having a 24-hour-generation time and various probabilities of any cell dying at any cell cycle. For a 53% cell death probability, no clones survive to 100 generations. For 52% cell death probability, 1 clone of 10 cells survives to 100 generations. For 51% cell death probability, 8 clones (average 19 cells per clone, largest clone 44 cells) survive to 100 generations, etc. Even for 45% cell death probability, there were only 127 (out of 500 initial clones) that reached the 100th generation (average 24,851 cells per clone, largest clone 89,851 cells).

*Calculations for p=0.44 were performed for 180 clones only, as the calculations required excessive computer time. For 180 clones, 54 reached 100 generations.

**For cell death rate zero, Monte Carlo and deterministic solutions are identical.
key words: precancer, precancerous condition, precancerous conditions, premalignant, preneoplastic, early cancer, early lesions, early lesion, neoplasm classification, cancer classification, in situ neoplasia, in situ cancer, non-invasive cancer, noninvasive cancer, pre-invasive cancer, preinvasive cancer, preneoplasia, premalignancy

Last modified: July 2, 2008

Biomedical Informatics, by Jules J. Berman, cover Perl Programming for Medicine and Biology, by Jules J. Berman, cover Ruby for Medicine and Biology, by Jules J. Berman, cover Perl: The Programming Language, by Jules J. Berman, cover