Positive feedback and angiogenesis in tumor growth control.

In vivo tumor growth data from experiments performed in our laboratory suggest that basic fibroblast growth factor (bFGF) and vascular endothelial growth factor (VEGF) are angiogenic signals emerging from an up-regulated genetic message in the proliferating rim of a solid tumor in response to tumor-wide hypoxia. If these signals are generated in response to unfavorable environmental conditions, i.e. a decrease in oxygen tension, then the tumor may play an active role in manipulating its own environment. We have idealized this type of adaptive behavior in our mathematical model via a parameter which represents the carrying capacity of the host for the tumor. If that model parameter is held constant, then environmental control is limited to tumor shape and mitogenic signal processing. However, if we assume that the response of the local stroma to these signals is an increase in the host's ability to support an ever larger tumor, then our models describe a positive feedback control system. In this paper, we generalize our previous results to a model including a carrying capacity which depends on the size of the proliferating compartment in the tumor. Specific functional forms for the carrying capacity are discussed. Stability criteria of the system and steady state conditions for these candidate functions are analyzed. The dynamics needed to generate stable tumor growth, including countervailing negative feedback signals, are discussed in detail with respect to both their mathematical and biological properties.

Host response in tumor growth and progression.

Tumor growth and progression result from complex controls that appear to be facilitated by the growth factors (GFs) which emerge from the tumor and find responsive targets both within the tumor and in the surrounding host. For example, basic fibroblast growth factor (bFGF) and vascular endothelial growth factor (VEGF) are both angiogenic signals which appear to emerge from upregulated genetic messages in the proliferating rim of a solid tumor in response to tumor-wide hypoxia. If these signals are generated in response to unfavorable environmental conditions, i.e. a tumor-wide decrease in oxygen tension, then the tumor may be playing a role in manipulating its own environment. Two questions are raised in this paper: (1) How does the host respond to such signals? (2) Is there a linkage between the host's response and the ultimate growth of the tumor? To answer these questions, we have idealized these adaptive signals within a mathematical model of tumor growth. The host response is characterized by a function which represents the host's carrying capacity for the tumor. If the function is constant, then environmental control is strictly limited to tumor shape and mitogenic signal processing. However, if we assume that the response of the local stroma to these signals is an increase in the host's ability to support an ever larger tumor, then the model describes a positive feedback controller. In this paper, we summarize our previous results and ask the question: What form of host response is reasonable, and how will it affect ultimate tumor growth? We examine some specific candidate response functions, and analyze them for system stability. In this model, unstable states correspond to 'infinite' tumor growth. We will also discuss countervailing negative feedback signals and their roles in maintaining tumor stability.

A theoretical explanation of "concomitant resistance".

Concomitant resistance is a tumor growth dynamic which results when the growth of a second tumor implant is inhibited by the presence of the first. Recently, we modeled tumor growth in the presence of a regenerating liver after partial hepatectomy (Michelson and Leith, Bull. Math. Biol. 57, 345-366, 1995), with an interlocking pair of growth control triads to account for the accelerated growth observed in both tissues. We also modeled tumor dormancy and recurrence as a dynamic equilibrium achieved between proliferating and quiescent subpopulations. In this paper those studies are extended to initially model the concomitant resistance case. Two interlocking model systems are proposed. In one an interactive competition between the tumor implants is described, while in the other purely proportional growth inhibition is described. The equilibria and dynamics of each system when the coefficients are held constant are presented for three subcases of model parameters. We show that the dynamic called concomitant resistance can be real or apparent, and that if the model coefficients are held constant, the only way to truly achieve concomitant resistance is by forcing one of the tumors into total quiescence. If this is the true state of the inhibited implant, then a non-constant recruitment signal is required to insure regrowth when the inhibitor mass is excised. We compare these theoretical results to a potential explanation of the phenomenon provided by Prehn (Cancer Res. 53, 3266-3269, 1993).

Secretion rates and levels of vascular endothelial growth factor in clone A or HCT-8 human colon tumour cells as a function of oxygen concentration.

Molecular and in situ hybridization studies have shown, in a number of cell types, that under hypoxic conditions, vascular endothelial growth factor (VEGF) mRNA expression is up-regulated and VEGF protein is concomitantly increased. To establish a quantitative relationship between VEGF protein levels and oxygenation, we exposed exponentially growing clone A or HCT-8 human colon tumour cells in vitro (22 h at 37 degrees C) to oxygen concentrations from 21% (air mixture) to 0.01%. Protein levels in cells and medium were then assayed using an enzyme-linked immunoabsorbent assay (ELISA). Intracellular levels of VEGF in clone A or HCT-8 cells exposed to either air (21% O2) or the 0.01% O2 mixture respectively increased from about 73 to 1270, and 1.5 to 1180 pg/10(6) cells (about 17- and 80-fold increases). The shapes of the response curves (log of the intracellular VEGF concentrations v. log oxygen concentration) for both cell types were sigmoidal. However, intracellular VEGF levels in HCT-8 cells were always less than that of clone A cells until levels of about 0.3 to 0.1% O2 were reached. Levels of VEGF in the supernatant were also increased after the 22 h hypoxic exposures. Because cell proliferation and clonogenicity were also measured, it was possible to estimate the secretion rates of VEGF for both cell lines as a function of oxygen percentage. For clone A cells, the secretion rate (pg/10(6) cells/h) in 21% O2 was 62.5. This rate increased to 428.8 pg/10(6) cells/h at 0.01% O2, a 7-fold increase. For HCT-8 cells, levels in the medium at 21% O2 were too low to be measured by ELISA. However, between 10% and 0.01% O2, secretion rates increased from 5.0 to 376.0 pg/10(6) cells/h, a 75-fold increase. Therefore, at very low O2 levels, VEGF secretion rates were similar in the two cell lines. We propose that the different VEGF responses of clone A and HCT-8 colon tumour cells to hypoxic stress in vitro are related to the in vivo observation that the respective hypoxic percentages of solid neoplasms originating from these cell lines are markedly different (i.e. about 3 versus 80%) at equivalent volumes of 750 mm3.

Comparison of basic fibroblast growth factor levels in clone A human colon cancer cells in vitro with levels in xenografted tumours.

We measured levels of basic fibroblast growth factor (FGF-2) in human colon cancer cells (clone A) in vitro and in xenografted solid tumours using a commercial enzyme-linked immunoassay. In Vitro, levels in unfed plateau phase or exponentially growing cells were low, averaging respectively about 2 and 8 pg 10(-6) cells. However, when solid tumours (average volumes 787 mm3) were cut into halves and either enzymatically disaggregated to obtain a cellular fraction or extracted in toto, levels were much higher. In the cellular fraction, values averaged 110 pg 10(-6) cells, while in whole tumour extracts, average values were 24 pg mg-1 tumour tissue. These results indicate that growth factor levels in solid neoplasms may differ markedly from those predicted from in vitro measurements. We hypothesise that the apparent increase in FGF-2 levels in vivo results primarily from the presence of a significant fraction of host cells (in particular, macrophages, which may contain high levels of FGF-2) within xenografted clone A neoplasms.

Levels of selected growth factors in viable and necrotic regions of xenografted HCT-8 human colon tumours.

Xenografted tumours were produced in nude mice by injection of HCT-8 human colon tumour cells. At average volumes of about 750 mm3, animals were injected with fast green vital dye, and 20 min later, tumours were excised and dissected into viable (stained) and necrotic portions (unstained). Viable and necrotic regions were then examined for cell yields, colony forming efficiencies, and levels of basic fibroblast growth factor (FGF-2), transforming growth factors-beta 1 and -alpha (TGF-beta 1, TGF-alpha), platelet derived growth factor (PDGF), and vascular endothelial growth factor (VEGF) using enzyme-linked immunoassay (ELISA) procedures. Levels in the viable and necrotic regions were compared to levels in unseparated tumours. The average extent of necrosis in HCT-8 tumours of this size was 64%. The data for cell yields, colony forming efficiencies FGF-2, VEGF, TGF-beta 1 and TGF-alpha indicated that values determined in the unseparated tumours could be understood on the basis of the weighted average between viable and necrotic tissue, with the higher values occurring in the viable tissue. Low levels of FGF-2 and VEGF were found in the necrotic portions of the tumour while no measurable levels of TGF-beta 1 and TGF-alpha could be determined. PDGF levels were, however, equivalent in both the viable and necrotic regions indicating that necrotic tissue could be an important reservoir for this growth factor.

Interlocking triads of growth control in tumors.

A pair of growth control triads are used to describe coincident tumor growth and liver regeneration after partial hepatectomy. The models are extensions of previous growth control models which describe tumor growth in an unperturbed host (Michelson and Leith, 1991, Bull. math. Biol. 53, 639-656; idem, 1992, Proceedings of the Third International Conference on Communications and Control, Vol. 2, pp. 481-490; idem, 1992 Bull. math. Biol. 55, 993-1011; idem, J. theor. Biol. 169, 327-338). The linkage between the two triads depends upon systemic signals carried by soluble factors, and mathematical descriptors based upon biological first principals are proposed. The sources of the growth factors, their targets and the processing of their signals are investigated. Analyses of equilibrium in the constant coefficients case and simulated growth curves for the dynamic system are presented, and the effects of growth factor-induced mitogenesis and angiogenesis are discussed in particular. A case is made for early and late responses in the coupled control system. The biology of the signal processing paradigm is placed within a new theoretical context and discussed with regard to tumor adaptation, liver differentiation and the development of a tumor hypoxic fraction.

Dormancy, regression, and recurrence: towards a unifying theory of tumor growth control.

In several recent publications, mathematical models of autocrine-paracrine and autocrine-paracrine-endocrine controls of growth in both homogeneous and heterogeneous tumor populations were developed (Michelson & Leith, 1991, Bull. math. Biol. 53, 639-656; 1992a, Proc. Third Int. Conf. Comm. Control, pp. 481-490; 1992b, Bull, math. Biol. 55, 993-1011). For the homogeneous case, a generic tumor was modeled as a single, growing population using the Verhulst equation of logistic growth. The heterogeneous tumor was modeled as a pair of populations, one proliferating and one quiescent. Mitogenic signals were represented as modifications to the Malthusian growth parameters, and adaptational signals were represented as modifications to the logistic carrying capacities. Interactions between populations were represented by competitive feedback and transition rates. In this paper a theory of growth control is proposed to determine whether tumor dormancy, regression, and recurrence can be explained by a more unifying theory of signal processing. The models developed earlier form the basis for this analysis. Dormancy is described as an equilibrium state from which tumors may re-emerge if that equilibrium is disrupted. The types of disruption in signal processing needed to induce recurrence are discussed with respect to surgery and wound healing. Based on this theory, it appears that sort of feedback between the host's ability to support the proliferating cells (adaptational signal processing) and the transition rates into and out of the proliferating and quiescent compartments must exist. A paradigm based on the development of hypoxia in a spherical tumor is proposed as that link.

Enhancement of thermal sensitivity of xenografted human DLD-2 tumors by administration of basic fibroblast growth factor.

It has previously been shown that administration of basic fibroblast growth factor (FGF-2) to mice bearing xenografted human DLD-2 carcinomas produces significant increases in tumor growth rates and decreased intratumor hypoxia, effects which appear to be secondary to changes in the vasculature. In this study, we treated DLD-2 tumors with FGF-2 (ip, 0.25 mg/kg, q.i.d. x 7) beginning on day 15 after implantation, when average tumor volumes were 238 mm3. One day after cessation of administration of FGF-2 (day 22 after implantation, average tumor volume 1748.1 mm3), clamped tumors were given hyperthermia (42.5 degrees C, 60 min) by water bath heating. The slower-growing tumors in the control mice (sham-injected with Hanks' basic salt solution) were clamped and subjected to hyperthermia treatment at equivalent average tumor volumes (1882.7 mm3), which occurred on day 26 after implantation. Tumors in control groups were clamped but not heated. The time needed for neoplasms to grow to twice their volumes at the time of hyperthermia treatment was 68 days for the FGF-2-treated neoplasms and 47 days for the controls, while 26 and 31 days were needed for the control groups which were not treated with heat. The relative growth delay induced by hyperthermia is therefore 16 (47-31) days for control neoplasms and 42 (68-26) days for FGF-2-treated tumors. Therefore, tumors in the mice injected with FGF-2 were significantly more sensitive to the hyperthermia than controls, by a factor of about 2.6 (42/16). This result indicates that administration of growth factors such as FGF-2 to mice bearing tumors may produce an increased sensitivity of the tumors to hyperthermia.

In vitro radiation sensitivity of the LNCaP prostatic tumor cell line.

Because there is extremely limited information on the intrinsic radiosensitivity of human prostatic cancer cells, we have investigated the in vitro radiation response of exponentially growing LNCaP cells. Due to the very poor colony-forming potential of the LNCaP cells, radiation survival was investigated using the dose-dependent (0-6 Gy) changes seen after X-irradiation in the shapes of regrowth curves. Survival was described using both the single-hit, multitarget (SHMT) equation and the linear-quadratic (LQ) equation. The values and 95% confidence limits of the extrapolation number (n), quasi-threshold dose (Dq), and mean lethal dose (D(o)) in SHMT terminology were respectively: 0.9 (0.7-1.0), 0.0 Gy, and 1.39 (0.11) Gy. The LQ alpha and beta parameters were respectively 6.80 (1.13) and -0.53 (2.89). The X-ray dose response of the LNCaP line is, therefore, purely exponential. The mean survival at the clinically relevant dose of 2 Gy (S2) was 51.2% for the LNCaP line. Comparison of the S2 value for the LNCaP line with previous investigations with other human prostatic cancer cell lines (DU145 and PC-3) indicates a mean S2 value of 47.6%, which suggests that human prostate cancer cells might lie toward the resistant side of the spectrum for various classes of human neoplasms.

Intrinsic and extrinsic characteristics of human tumors relevant to radiosurgery: comparative cellular radiosensitivity and hypoxic percentages.

UNLABELLED: We have collected the in vitro x-ray radiation survival characteristics of 181 lines from 12 different classes of exponentially growing human tumor cells (sarcomas, lung cancers, colo-rectal cancers, medulloblastomas, melanoma, breast cancers, prostate cancers, renal cell cancers, grades III and IV brain tumors, ovarian, and head and neck cancers). This information was used to intercompare survival after single high doses of 20-40 Gy for each tumor line. Radiosensitivities could roughly be divided into two groups. The more radiosensitive group included: sarcoma, small-cell lung cancer, non-small cell lung cancer, colorectal cancer, medulloblastoma and melanoma. The more radioresistant group included breast, prostate, renal cell, primary brain tumors, ovarian tumors, and head and neck cancers. Using a model of a 3 cm diameter brain lesion containing about 1.4 x 10(9) oxic cells, the single doses calculated to reduce survival to 1 cell were: sarcoma and small cell lung cancers-22-23 Gy; melanoma-25 Gy; non-small cell lung and colorectal cancer-26 Gy; medullo-blastoma-28 Gy; breast, prostate, renal cell, primary brain tumors, ovarian tumors, and head and neck cancers-30-36 Gy. If, however, tumors contained on average 20 percent hypoxic cells, the dose needed for equivalent cell killing increased by about a factor of 2.6-2.8. Also, there was no correlation between the ranking of relative radiosensitivities of the various classes of tumor cells at high doses (as in radiosurgery) to the sensitivity at low doses (as in conventional fractionated radiotherapy). CONCLUSION: available information on the intrinsic radiosensitivity of human tumor cells indicates that meaningful differences exist among different histological classes of neoplasm that are relevant to the single high doses used in radioneurosurgery, and which could constitute a basis for "tailoring" the administered dose to the particular neoplasm. However, if intracerebral lesions contain a large number of hypoxic cells (e.g., 20%), this may constitute a significant problem.

Effects of administration of basic fibroblast growth factor on hypoxic fractions in xenografted DLD-2 human tumours: time dependence.

A previous publication (Leith et al., 1992) showed that administration of basic fibroblast growth factor (FGF-2, 0.25 mg kg-1, q.i.d. x 7) to mice bearing xenografted DLD-2 human colon cancers would increase treated tumour growth rates as compared to control neoplasms. Additionally, at the end of the 7 day treatment period, clonogenic excision assays showed that the percentage of hypoxic cells in tumours from mice receiving FGF-2 administration was significantly decreased as compared to control neoplasms (from about 42 to about 19%). The present study was undertaken to better define the kinetics of changes in hypoxic percentages as a function of tumour volume and FGF-2 treatment. In sham-injected control tumours, the hypoxic percentage increased from about 14% at day 15 postimplantation, (i.e. when sham- or FGF-2 injections were begun) to about 42% by day 22, and to about 75% at 29 days postimplantation (respective average volumes 220, 910, and 2810 mm3). In contrast, the hypoxic percentages in mice treated with FGF-2 remained at the levels seen in control mice on day 15, not only throughout the 7 day FGF-2 treatment schedule, but for at least 1 week after the cessation of growth factor administration. The hypoxic percentage was 16% on day 29 postimplantation, even though average tumour volumes were about 4325 mm3. These data show that the effect of FGF-2 administration on tumour growth rate and hypoxic percentages in xenografted DLD-2 neoplasms is rapid, and continues for some period of time even after administration is ended. Studies of tumour perfusion with injected 86RbCl at equivalent tumour volumes of about 1800 mm3 indicated that the percentage of cardiac output to FGF-2 treated tumours was 33% greater than in sham-injected control neoplasms.

Growth factors and growth control of heterogeneous cell populations.

In an earlier work a model of the autocrine and paracrine pathways of tumor growth control was developed (Michelson and Leith. 1991. Autocrine and paracrine growth factors in tumor growth. Bull. math. Biol. 53, 639-656). The target population, a generic tumor, was modeled as a single, homogeneous population using the standard Verhulst equation of logistic growth. Mitogenic signals were represented by modifications to the Malthusian growth parameter and adaptational signals were represented by modifications to the carrying capacity. Three growth scenarios were described: (1) normal tissue wound healing, (2) unperturbed tumor growth, and (3) tumor growth in a radiation damaged environment, a phenomenon termed the Tumor Bed Effect (TBE). In this paper, we extend those results to include a "triad" of growth factor controls (autocrine, paracrine and endocrine) and heterogeneity of the target population. The heterogeneous factors in the model represent either intrinsic, epigenetic or environmental differences in both normally differentiating tissues and tumors. Three types of growth are modeled: (1) normal tissue differentiation or wound healing, assuming no communication between differentiated and undifferentiated cell compartments; (2) normal wound healing with feedback inhibition, due to signalling from the differentiated compartment; and (3) the development of hypoxia in a spherical tumor. The signal processing within the triad is discussed for each model and biologically reasonable constraints are defined for limits on growth control.