J. theor. Biol. (1991) 152, 377-403
Analysis of the Roles of Microvessel Endothelial Cell Random Motility and Chemotaxis in Angiogenesis CYNTHIA L. STOKESt AND DOUGLAS A. LAUFFENBURGER~
Department of Chemical Engineering, University of Pennsylvania, Philadelphia, PA 19104, U.S.A. (Received on 24 August 1990, Accepted on 5 March 1991) The growth of new capillary blood vessels, or angiogenesis, is a prominent component of numerous physiological and pathological conditions. An understanding of the co-ordination of underlying cellular behaviors would be helpful for therapeutic manipulation of the process. A probabilistic mathematical model of angiogenesis is developed based upon specific microvessel endothelial cell (MEC) functions involved in vessel growth. The model focuses on the roles of MEC random motility and chemotaxis, to test the hypothesis that these MEC behaviors are of critical importance in determining capillary growth rate and network structure. Model predictions are computer simulations of microvessel networks, from which questions of interest are examined both qualitatively and quantitatively. Results indicate that a moderate MEC chemotactic response toward an angiogenic stimulus, similar to that measured in vitro in response to acidic fibroblast growth factor, is necessary to provide directed vascular network growth. Persistent random motility alone, with initial budding biased toward the stimulus, does not adequately provide directed network growth. A significant degree of randomness in cell migration direction, however, is required for vessel anastomosis and capillary loop formation, as simulations with an overly strong chemotactic response produce network structures largely absent of these features. The predicted vessel extension rate and network structure in the simulations are quantitatively consistent with experimental observations of angiogenesis in rive. This suggests that the rate of vessel outgrowth is primarily determined by MEC migration rate, and consequently that quantitative in vitro migration assays might be useful tools for the prescreening of possible angiogenesis activators and inhibitors. Finally, reduction of MEC speed results in substantial inhibition of simulated angiogenesis. Together, these results predict that both random motility and chemotaxis are MEC functions critically involved in determining the rate and morphology of new microvessel network growth.
Introduction
Angiogenesis, the growth and d e v e l o p m e n t o f new capillary b l o o d vessels, occurs in n u m e r o u s physiological and pathological conditions, including e m b r y o n i c development, w o u n d healing, growth o f solid tumors, a n d d e v e l o p m e n t of atherosclerotic plaques ( F o l k m a n & Klagsbrun, 1987). One a p p r o a c h to the study o f angiogenesis is to investigate the underlying cellular behaviors which comprise t Author to whom correspondence should be addressed. Present address: Department of Chemical Engineering, University of Houston, Houston, TX 77204-4792, U.S.A. Present address: Department of Chemical Engineering, University of Illinois at Urbana-Champaign, 297 Roger Adams Laboratory, 1209 W. California Street, Urbana, IL 61801, U.S.A. 377 0022-5193/91/190377+27 $03.00/0 © 1991 Academic Press Limited
378
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the process. Experiments have demonstrated that the microvessel endothelial cell (MEC) functions involved in neovascularization are, at a minimum, cell proliferation (Ausprunk & Folkman, 1977; Sholley et al., 1984), migration (Ausprunk & Folkman, 1977; Clark & Clark, 1939), protease production for the degradation of basement membrane and extracellular matrix (Glaser et al., 1983; Gross et al., 1983; Moscatelli et al., 1985), and basement membrane production and deposition (Ingber & Folkman, 1988; Maragoudakis et al., 1988). However, utilization of this information requires an understanding of the mechanistic relationships by which these functions are co-ordinated. In addition, quantitative investigations are needed to determine the relative contribution of the various functions in governing microvessel network growth rate and structure. To aid in progress toward these goals, we present a mathematical model of angiogenesis based on a postulated co-ordination of MEC functions. Because experimental data (Rupnick et al., 1988; Stokes et al., 1990a) suggest that the rate of in vitro MEC motility is comparable to that of angiogenesis in vivo, and in vivo observations show that the growth of vessels is quite directed towards an angiogenic stimulus, our model has been designed specifically to explore how MEC motility characteristics affect the rate of growth and morphology of developing microvessel networks. The model is used to investigate the effects of both random motility, characterized by speed and persistence time, and chemotaxis, described by the chemotactic responsiveness. Our model uses a structured, probabilistic framework capable of simulating the development of individual microvessels and resulting networks. Computational results consist of "theoretical pictures" of network structures as they evolve transiently, which can be analyzed for quantitative data such as vessel lengths and mean network expansion rates. Thus, both qualitative and quantitative results can be compared to experimental observations in the literature to judge the hypotheses of the model. In contrast, previous models of neovascularization have used a continuum, deterministic framework (Balding & McElwain, 1985; Liotta et al., 1977; Zawicki et al., 1981). These models predicted averaged quantities such as vessel and tip densities per volume tissue. Though reasonably successful in exhibiting correct trends for the particular in vivo systems under consideration, they were only partially based on underlying cellular or biochemical processes so that model parameters could not be measured independently. Therefore, these models possessed limited predictive value for understanding the relationship between cell functions and angiogenesis. Moreover, the continuum formulation in terms of averaged quantities allowed quantitation of network expansion rates but not detailed features such as vessel lengths and distances between buds or anastomoses. Neither could they analyze information regarding network structure, or morphology. Because microcirculatory blood flow characteristics are determined by capillary loop characteristics, and because directional growth of vessels is manifested in the morphology, information on network structure may be the most discriminatory type of data which can be predicted for comparing models and mechanistic hypotheses to experimental observations. In our work, parameter values for MEC random motility, chemotaxis, proliferation, and vessel budding probabilities are obtained independently by direct
MOTILITY
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IN
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measurement or estimation from experimental observations, rather than adjusted to give a desired result. This enables us to investigate a number of questions regarding the roles of MEC migration and proliferation behavior. These questions include whether a MEC chemotactic response is necessary to result in the directional development of microvessel networks commonly observed in vivo. If so, is the magnitude of the necessary chemotactic response on the same order as that we have measured in the laboratory (Stokes et al., 1990a)? As an alternative mechanism for directed growth, one might conjecture that the highly-persistent random motility typical of tissue cells (Glasgow et al., 1989; Stokes et al., 1991) could cause directed growth of vessels in vivo if initial vessel budding was biased toward the angiogenic stimulus. Experimental observations indicating that vessels sprout in the direction of an attractant (Eddy & Casarett, 1973; Wolff et al., 1985) motivate this alternative. Another question is whether there might be an optimal level of chemotaxis for effective microvessel network formation, given that our laboratory measurements yield a significant but only moderate chemotactic response (Stokes et al., 1990a). We have also addressed questions regarding the effects of MEC speed and persistence time on network characteristics. In particular, is the rate of vessel outgrowth primarily influenced by MEC migration rate, and is the inhibition of random motility sufficient to substantially repress angiogenesis despite a normal level ofchemotactic directional bias? We have previously found that the combination of/3-cyclodextdn tetradecasulfate and hydrocortisone dramatically inhibits MEC random motility (Stokes et ai., 1990b), correlating with its suppression of angiogenesis in vivo (Folkman et al., 1989). Finally, the effects of the rate of cell proliferation were studied to investigate whether proliferation rate governs the rate of vessel growth.
Materials and Methods MATHEMATICAL
MODEL
The sequence of events in angiogenesis which the model emulates can be described as follows: In response to a nearby stimulus, MEC in the pre-existing vessels begin to proliferate (Sholley et al., 1984), and the basement membrane surrounding these vessels is degraded in apparently random areas (Ausprunk & Folkman, 1977). MEC begin to migrate out of existing vessels into the tissue space, with several cells lining up to form solid sprouts. Sholley et al. (1984) demonstrated that MEC can be "redistributed" among sprouts, moving from one sprout into another. This allowed significant vessel growth even when cell proliferation was prevented. The MEC in a sprout proliferate, producing more cells for the continued elongation of the sprout. Several investigators (Ausprunk & Folkman, 1977; Sholley et al., 1984) have observed that most MEC proliferation occurs just behind the leading edge of the expanding microvascular bed. The vessel lumen begins to form early in the formation of the sprout. Sprouts run into one another and anastomose to form loops. More sprouts bud of[ of newly established vessels, providing the mechanism by which the microvascular network continues to enlarge. Some new sprouts elongate only for a short distance and then regress, either by reverse migration or by the pinching off of the
380
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AND
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sprout and apparent disintegration of the MEC (Clark & Clark, 1939; van den Brenk, 1955; Zawicki et al., 1981). The new vessels deposit basement membrane (Ingber & Folkman, 1988), and blood flow is established throughout the new vasculature as it develops. Finally, pericytes take up positions along the abluminal surface of the capillaries. We seek to include in the model only the most fundamental events necessary to form a new microvessel network. The model starts with vessels budding off of pre-existing, or "parent", vessels, includes cell proliferation, migration (both random and chemotaxis), anastomosis, and budding of new sprouts off newly formed vessels. The rate of degradation of basement membrane and extracellular matrix is not explicitly involved, but rather is assumed to be included in rates given for budding and migration. All events which occur once a sprout segment is formed (lumen formation, basement membrane production and deposition, and pericyte incorporation) are ignored since they do not affect the actual formation of the network. The regression of capillaries is not included for the sake of simplicity. The mathematical model is stochastic in its description of the MEC migration as well as its description of vessel branching. A stochastic (probabilistic) mathematical description of the cell motility was needed because of the random walk characteristics of cell motility (Dunn, 1981). The model is confined to two dimensions, which should be a reasonable approximation of the vessel growth which takes place in thin ( P~in was further migration o f the tip cell allowed and thereby further elongation o f the entire sprout. Examination o f the generated values o f density and velocity for numerous simulations with k s = 0-02 hr -~ and kb = S/100 hr -~ [with various values o f S (10-80 i~m hr-~), Pv(1-18 hr), and Kao (0-4800 ~m 2 hr-2)] did not show vessel densities decreasing to the minimum during the simulation. This indicates that the proliferation and redistribution rates are able to provide cells quickly enough to allow unlimited motility and vessel elongation. The redistribution acts primarily to provide cells at the early stages o f a sprout's growth (before it has many cells to proliferate), while proliferation provides most cells at later stages. Thus, the model predicts that as long as normal cell proliferation is allowed, proliferation does not limit the rate of vessel growth. The rate of redistribution acts similarly when cell proliferation is prevented (kg = 0 ; e.g. Fig. 2), providing cells at a sufficient rate to allow relatively uninterrupted vessel elongation. In these latter simulations, "stop-and-start" growth was sometimes observed. EFFECTS OF CHEMOTAXIS Figure 3 illustrates a simulation for the base case: only random migration (Kao = 0), S = 40 p.m hr -~, Pv = 3 hr, and all other parameter values as shown in Table 1. The initial velocity (speed and direction) of the sprouts was random. This simulation and others not shown reveal that purely random motility with random initial direction of growth does not result in networks similar to those observed in vivo. Most significant to this conclusion is that the directionality of vessel growth is haphazard, which does not resemble the directed vessel growth observed in vivo. Because of this disagreement, it is not useful to make any quantitative comparisons to in vivo networks. We found that until the value o f the chemotactic responsiveness was increased to about 8 -- 1, the directional growth o f vessels was not significantly altered from that shown in Fig. 3 for 8 = 0. Figures 4 and 5 illustrate the results when higher levels o f chemotaxis were incorporated in the model. Again, S = 40 I~m hr -~, Pv = 3 hr, and the initial directions o f sprout growth were random. For 8 around 1-0 or greater (Fig. 4 with 8 = 1.5, or Kao= 2400 ixm2 hr-2), most of the growth is in the general direction of the attractant source, similar to angiogenesis in vivo. In this simulation, the average network expansion rate for the simulation was 0-32 mm day -l, and the average of 12 simulations with 8 = 1.5 was 0-26 mm day -~. For a slightly lower chemotactic responsiveness ( 8 = 1-0), the average network expansion rate for 12 simulations was 0.17 mm day -1. In vivo measurements o f this quantity are in the range of 0.09-0.3 mm day -1 (Clark & Clark, 1939; Van Den Brenk, 1955; Zawicki et al., 1981) so the simulations with reasonable levels of chemotaxis are in agreement. In addition, the average vessel length between branches was 0.15 mm for the 12 simulations at both 8 = 1 and 1-5. This is also similar to in vivo networks, where most vessels are between 0.1 and 0.3 mm long (van den Brenk, 1955; Zawicki et al., 1981 ). Thus, with in vitro migration and proliferation characteristics incorporated, networks can be simulated which compare well to in vivo microvessel networks both qualitatively and quantitatively.
MOTILITY
AND
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391
IN ANGIOGENESIS
1'5
1.0-
0-o
_
-0.5
-I.0
-I-5 -0-5
I 0.O
t 0.:5
t I-0
I 1.5
1 2.0
2.5
FIG. 3. Angiogenesis simulation with r a n d o m motility only (no chemotaxis, Ka0 = 0). S = 40 ixm hr -~, P. = 3 hr, a n d other parameter values are given in Table 1. Initial direction o f growth of all sprouts was r a n d o m . Axis units are in millimeters, a n d the pre-existing vasculature is represented by the horizontal line.
However, for the level of chemotaxis with 8 = 1-1.5, a certain amount of vessel growth away from the attractant source is predicted in some simulations. Most experimental observations do not report such growth, though its observation might be difficult because of pre-existing vasculature. The discrepancy might occur because all budding in the simulations was in random directions, while experimental observations indicate that vessels generally bud in the direction the attractant source (Eddy & Casarett, 1973; Wolff et aL, 1985). When that restriction was incorporated into the calculations, vessel growth in the direction away from the attractant source was nearly completely eliminated (not shown). The mechanism for directional budding in vivo is probably a combination of protease production stimulated by attractant, which may be greater on the side of a vessel closer to the attractant, as well as mechanical factors due to buckling, as proposed by Waxman (1981). The simulation in Fig. 5, where 8 = 5.0 (Ka0 = 8000 ixm2 hr-2), illustrates the result with a strong chemotactic responsiveness, roughly a factor of 2 to 3 greater than expected from the level measured in vitro in response to aFGF (Stokes et al., 1990a). In these simulations, the vessels are somewhat longer, straighter, and have fewer
392
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2.0
t-5
1,0
0-.5
0,0
-0.5
-I.0 -0-.5
I 0-0
I 0.5
I 1.0
I I.,5
I 2.0
2.5
FIG. 4. Angiogenesis simulation with moderate chemotactic responsiveness (,5= 1.5, or t
Analysis of the Roles of Microvessel Endothelial Cell Random Motility and Chemotaxis in Angiogenesis CYNTHIA L. STOKESt AND DOUGLAS A. LAUFFENBURGER~
Department of Chemical Engineering, University of Pennsylvania, Philadelphia, PA 19104, U.S.A. (Received on 24 August 1990, Accepted on 5 March 1991) The growth of new capillary blood vessels, or angiogenesis, is a prominent component of numerous physiological and pathological conditions. An understanding of the co-ordination of underlying cellular behaviors would be helpful for therapeutic manipulation of the process. A probabilistic mathematical model of angiogenesis is developed based upon specific microvessel endothelial cell (MEC) functions involved in vessel growth. The model focuses on the roles of MEC random motility and chemotaxis, to test the hypothesis that these MEC behaviors are of critical importance in determining capillary growth rate and network structure. Model predictions are computer simulations of microvessel networks, from which questions of interest are examined both qualitatively and quantitatively. Results indicate that a moderate MEC chemotactic response toward an angiogenic stimulus, similar to that measured in vitro in response to acidic fibroblast growth factor, is necessary to provide directed vascular network growth. Persistent random motility alone, with initial budding biased toward the stimulus, does not adequately provide directed network growth. A significant degree of randomness in cell migration direction, however, is required for vessel anastomosis and capillary loop formation, as simulations with an overly strong chemotactic response produce network structures largely absent of these features. The predicted vessel extension rate and network structure in the simulations are quantitatively consistent with experimental observations of angiogenesis in rive. This suggests that the rate of vessel outgrowth is primarily determined by MEC migration rate, and consequently that quantitative in vitro migration assays might be useful tools for the prescreening of possible angiogenesis activators and inhibitors. Finally, reduction of MEC speed results in substantial inhibition of simulated angiogenesis. Together, these results predict that both random motility and chemotaxis are MEC functions critically involved in determining the rate and morphology of new microvessel network growth.
Introduction
Angiogenesis, the growth and d e v e l o p m e n t o f new capillary b l o o d vessels, occurs in n u m e r o u s physiological and pathological conditions, including e m b r y o n i c development, w o u n d healing, growth o f solid tumors, a n d d e v e l o p m e n t of atherosclerotic plaques ( F o l k m a n & Klagsbrun, 1987). One a p p r o a c h to the study o f angiogenesis is to investigate the underlying cellular behaviors which comprise t Author to whom correspondence should be addressed. Present address: Department of Chemical Engineering, University of Houston, Houston, TX 77204-4792, U.S.A. Present address: Department of Chemical Engineering, University of Illinois at Urbana-Champaign, 297 Roger Adams Laboratory, 1209 W. California Street, Urbana, IL 61801, U.S.A. 377 0022-5193/91/190377+27 $03.00/0 © 1991 Academic Press Limited
378
c . L . STOKES AND D, A, LAUFFENBURGER
the process. Experiments have demonstrated that the microvessel endothelial cell (MEC) functions involved in neovascularization are, at a minimum, cell proliferation (Ausprunk & Folkman, 1977; Sholley et al., 1984), migration (Ausprunk & Folkman, 1977; Clark & Clark, 1939), protease production for the degradation of basement membrane and extracellular matrix (Glaser et al., 1983; Gross et al., 1983; Moscatelli et al., 1985), and basement membrane production and deposition (Ingber & Folkman, 1988; Maragoudakis et al., 1988). However, utilization of this information requires an understanding of the mechanistic relationships by which these functions are co-ordinated. In addition, quantitative investigations are needed to determine the relative contribution of the various functions in governing microvessel network growth rate and structure. To aid in progress toward these goals, we present a mathematical model of angiogenesis based on a postulated co-ordination of MEC functions. Because experimental data (Rupnick et al., 1988; Stokes et al., 1990a) suggest that the rate of in vitro MEC motility is comparable to that of angiogenesis in vivo, and in vivo observations show that the growth of vessels is quite directed towards an angiogenic stimulus, our model has been designed specifically to explore how MEC motility characteristics affect the rate of growth and morphology of developing microvessel networks. The model is used to investigate the effects of both random motility, characterized by speed and persistence time, and chemotaxis, described by the chemotactic responsiveness. Our model uses a structured, probabilistic framework capable of simulating the development of individual microvessels and resulting networks. Computational results consist of "theoretical pictures" of network structures as they evolve transiently, which can be analyzed for quantitative data such as vessel lengths and mean network expansion rates. Thus, both qualitative and quantitative results can be compared to experimental observations in the literature to judge the hypotheses of the model. In contrast, previous models of neovascularization have used a continuum, deterministic framework (Balding & McElwain, 1985; Liotta et al., 1977; Zawicki et al., 1981). These models predicted averaged quantities such as vessel and tip densities per volume tissue. Though reasonably successful in exhibiting correct trends for the particular in vivo systems under consideration, they were only partially based on underlying cellular or biochemical processes so that model parameters could not be measured independently. Therefore, these models possessed limited predictive value for understanding the relationship between cell functions and angiogenesis. Moreover, the continuum formulation in terms of averaged quantities allowed quantitation of network expansion rates but not detailed features such as vessel lengths and distances between buds or anastomoses. Neither could they analyze information regarding network structure, or morphology. Because microcirculatory blood flow characteristics are determined by capillary loop characteristics, and because directional growth of vessels is manifested in the morphology, information on network structure may be the most discriminatory type of data which can be predicted for comparing models and mechanistic hypotheses to experimental observations. In our work, parameter values for MEC random motility, chemotaxis, proliferation, and vessel budding probabilities are obtained independently by direct
MOTILITY
AND
CHEMOTAXIS
IN
AN(3IOGENESIS
379
measurement or estimation from experimental observations, rather than adjusted to give a desired result. This enables us to investigate a number of questions regarding the roles of MEC migration and proliferation behavior. These questions include whether a MEC chemotactic response is necessary to result in the directional development of microvessel networks commonly observed in vivo. If so, is the magnitude of the necessary chemotactic response on the same order as that we have measured in the laboratory (Stokes et al., 1990a)? As an alternative mechanism for directed growth, one might conjecture that the highly-persistent random motility typical of tissue cells (Glasgow et al., 1989; Stokes et al., 1991) could cause directed growth of vessels in vivo if initial vessel budding was biased toward the angiogenic stimulus. Experimental observations indicating that vessels sprout in the direction of an attractant (Eddy & Casarett, 1973; Wolff et al., 1985) motivate this alternative. Another question is whether there might be an optimal level of chemotaxis for effective microvessel network formation, given that our laboratory measurements yield a significant but only moderate chemotactic response (Stokes et al., 1990a). We have also addressed questions regarding the effects of MEC speed and persistence time on network characteristics. In particular, is the rate of vessel outgrowth primarily influenced by MEC migration rate, and is the inhibition of random motility sufficient to substantially repress angiogenesis despite a normal level ofchemotactic directional bias? We have previously found that the combination of/3-cyclodextdn tetradecasulfate and hydrocortisone dramatically inhibits MEC random motility (Stokes et ai., 1990b), correlating with its suppression of angiogenesis in vivo (Folkman et al., 1989). Finally, the effects of the rate of cell proliferation were studied to investigate whether proliferation rate governs the rate of vessel growth.
Materials and Methods MATHEMATICAL
MODEL
The sequence of events in angiogenesis which the model emulates can be described as follows: In response to a nearby stimulus, MEC in the pre-existing vessels begin to proliferate (Sholley et al., 1984), and the basement membrane surrounding these vessels is degraded in apparently random areas (Ausprunk & Folkman, 1977). MEC begin to migrate out of existing vessels into the tissue space, with several cells lining up to form solid sprouts. Sholley et al. (1984) demonstrated that MEC can be "redistributed" among sprouts, moving from one sprout into another. This allowed significant vessel growth even when cell proliferation was prevented. The MEC in a sprout proliferate, producing more cells for the continued elongation of the sprout. Several investigators (Ausprunk & Folkman, 1977; Sholley et al., 1984) have observed that most MEC proliferation occurs just behind the leading edge of the expanding microvascular bed. The vessel lumen begins to form early in the formation of the sprout. Sprouts run into one another and anastomose to form loops. More sprouts bud of[ of newly established vessels, providing the mechanism by which the microvascular network continues to enlarge. Some new sprouts elongate only for a short distance and then regress, either by reverse migration or by the pinching off of the
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D. A. LAUFFENBURGER
sprout and apparent disintegration of the MEC (Clark & Clark, 1939; van den Brenk, 1955; Zawicki et al., 1981). The new vessels deposit basement membrane (Ingber & Folkman, 1988), and blood flow is established throughout the new vasculature as it develops. Finally, pericytes take up positions along the abluminal surface of the capillaries. We seek to include in the model only the most fundamental events necessary to form a new microvessel network. The model starts with vessels budding off of pre-existing, or "parent", vessels, includes cell proliferation, migration (both random and chemotaxis), anastomosis, and budding of new sprouts off newly formed vessels. The rate of degradation of basement membrane and extracellular matrix is not explicitly involved, but rather is assumed to be included in rates given for budding and migration. All events which occur once a sprout segment is formed (lumen formation, basement membrane production and deposition, and pericyte incorporation) are ignored since they do not affect the actual formation of the network. The regression of capillaries is not included for the sake of simplicity. The mathematical model is stochastic in its description of the MEC migration as well as its description of vessel branching. A stochastic (probabilistic) mathematical description of the cell motility was needed because of the random walk characteristics of cell motility (Dunn, 1981). The model is confined to two dimensions, which should be a reasonable approximation of the vessel growth which takes place in thin ( P~in was further migration o f the tip cell allowed and thereby further elongation o f the entire sprout. Examination o f the generated values o f density and velocity for numerous simulations with k s = 0-02 hr -~ and kb = S/100 hr -~ [with various values o f S (10-80 i~m hr-~), Pv(1-18 hr), and Kao (0-4800 ~m 2 hr-2)] did not show vessel densities decreasing to the minimum during the simulation. This indicates that the proliferation and redistribution rates are able to provide cells quickly enough to allow unlimited motility and vessel elongation. The redistribution acts primarily to provide cells at the early stages o f a sprout's growth (before it has many cells to proliferate), while proliferation provides most cells at later stages. Thus, the model predicts that as long as normal cell proliferation is allowed, proliferation does not limit the rate of vessel growth. The rate of redistribution acts similarly when cell proliferation is prevented (kg = 0 ; e.g. Fig. 2), providing cells at a sufficient rate to allow relatively uninterrupted vessel elongation. In these latter simulations, "stop-and-start" growth was sometimes observed. EFFECTS OF CHEMOTAXIS Figure 3 illustrates a simulation for the base case: only random migration (Kao = 0), S = 40 p.m hr -~, Pv = 3 hr, and all other parameter values as shown in Table 1. The initial velocity (speed and direction) of the sprouts was random. This simulation and others not shown reveal that purely random motility with random initial direction of growth does not result in networks similar to those observed in vivo. Most significant to this conclusion is that the directionality of vessel growth is haphazard, which does not resemble the directed vessel growth observed in vivo. Because of this disagreement, it is not useful to make any quantitative comparisons to in vivo networks. We found that until the value o f the chemotactic responsiveness was increased to about 8 -- 1, the directional growth o f vessels was not significantly altered from that shown in Fig. 3 for 8 = 0. Figures 4 and 5 illustrate the results when higher levels o f chemotaxis were incorporated in the model. Again, S = 40 I~m hr -~, Pv = 3 hr, and the initial directions o f sprout growth were random. For 8 around 1-0 or greater (Fig. 4 with 8 = 1.5, or Kao= 2400 ixm2 hr-2), most of the growth is in the general direction of the attractant source, similar to angiogenesis in vivo. In this simulation, the average network expansion rate for the simulation was 0-32 mm day -l, and the average of 12 simulations with 8 = 1.5 was 0-26 mm day -~. For a slightly lower chemotactic responsiveness ( 8 = 1-0), the average network expansion rate for 12 simulations was 0.17 mm day -1. In vivo measurements o f this quantity are in the range of 0.09-0.3 mm day -1 (Clark & Clark, 1939; Van Den Brenk, 1955; Zawicki et al., 1981) so the simulations with reasonable levels of chemotaxis are in agreement. In addition, the average vessel length between branches was 0.15 mm for the 12 simulations at both 8 = 1 and 1-5. This is also similar to in vivo networks, where most vessels are between 0.1 and 0.3 mm long (van den Brenk, 1955; Zawicki et al., 1981 ). Thus, with in vitro migration and proliferation characteristics incorporated, networks can be simulated which compare well to in vivo microvessel networks both qualitatively and quantitatively.
MOTILITY
AND
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391
IN ANGIOGENESIS
1'5
1.0-
0-o
_
-0.5
-I.0
-I-5 -0-5
I 0.O
t 0.:5
t I-0
I 1.5
1 2.0
2.5
FIG. 3. Angiogenesis simulation with r a n d o m motility only (no chemotaxis, Ka0 = 0). S = 40 ixm hr -~, P. = 3 hr, a n d other parameter values are given in Table 1. Initial direction o f growth of all sprouts was r a n d o m . Axis units are in millimeters, a n d the pre-existing vasculature is represented by the horizontal line.
However, for the level of chemotaxis with 8 = 1-1.5, a certain amount of vessel growth away from the attractant source is predicted in some simulations. Most experimental observations do not report such growth, though its observation might be difficult because of pre-existing vasculature. The discrepancy might occur because all budding in the simulations was in random directions, while experimental observations indicate that vessels generally bud in the direction the attractant source (Eddy & Casarett, 1973; Wolff et aL, 1985). When that restriction was incorporated into the calculations, vessel growth in the direction away from the attractant source was nearly completely eliminated (not shown). The mechanism for directional budding in vivo is probably a combination of protease production stimulated by attractant, which may be greater on the side of a vessel closer to the attractant, as well as mechanical factors due to buckling, as proposed by Waxman (1981). The simulation in Fig. 5, where 8 = 5.0 (Ka0 = 8000 ixm2 hr-2), illustrates the result with a strong chemotactic responsiveness, roughly a factor of 2 to 3 greater than expected from the level measured in vitro in response to aFGF (Stokes et al., 1990a). In these simulations, the vessels are somewhat longer, straighter, and have fewer
392
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A N D D. A. L A U F F E N B U R G E R
2.0
t-5
1,0
0-.5
0,0
-0.5
-I.0 -0-.5
I 0-0
I 0.5
I 1.0
I I.,5
I 2.0
2.5
FIG. 4. Angiogenesis simulation with moderate chemotactic responsiveness (,5= 1.5, or t
