SESSION 6

MASS TRANSFER - II

Chairman: R. Collins

ABSTRACT

We present results of numerical simulations of a mathematical model simulating mass transfer in the development of a tumor, resulting in its encapsulation and lobulation. Important differences between one dimensional Cartesian, cylindrical, and spherical geometries are reported.

INTRODUCTION

A good understanding of tumor evolution is essential for the development of efficient prevention and cure. Tumors, whether benign or malignant, are composed of two basic components: (1) cancerous cells that constitute its parenchyma, and (2) supportive stroma made of connective tissue and blood vessels. Benign tumors are often contained in a dense band (or capsule) comprised of compressed connective tissue. Encapsulated benign tumors are either a continuum of tumor cells, or appear as several lobes of different sizes, separated by a small amount of intervening connective tissue stroma, all contained in a larger capsule. Cells of a malignant tumor, on the other hand, are not restrained by a capsule. Development of artificial ways of inducing the encapsulation process, and hence containing the spread of malignant tumor cells is desirable.^1-3

Mathematical modeling of tumor growth has, in general, been restricted to simple models.^4-6 Evaluation of tumor growth models was reported by Vaidya and Alexandro⁷. Recent developments in the understanding of the convection-diffusion and reaction-diffusion equations,^8,9 their application to mathematical biology,^l0 and progress in numerical schemes to solve these equations,^ll has led to modeling of tumor growth as a reaction-diffusion process.^l2-13 Perumpanani et al.¹³ developed and numerically solved two mathematical models in one-dimensional Cartesian geometry to explain the formation of the capsule and the multi-lobular nature of some tumors. In this study we extend the model used by Perumpanani et a1.¹³ in two specific ways. First we generalize the tumor cell generation term to simulate more complex growth than is possible by the growth model used by them. Second, we carry out the simulations in one-dimensional Cartesian, as well as cylindrical and spherical geometries. In certain cases, as is shown below, the difference between the results for different geometries can be quite significant.

MODEL

The partial differential equations governing the balance of the tumor cells m(r,t), and connective tissue c(r,t) are¹³

where g(m,c) is the generation term; b is a constant; n is 0, 1 or 2 for Cartesian, cylindrical and spherical geometries, respectively. Note that the convective tissue does not diffuse-it is only convected by the diffusing tumor cells. The connective tissue flux is proportional to the connective tissue concentration c(r,t) and the tumor cell flux J_m. Parameter k is a key parameter in the model that represents the ability of the diffusing tumor cells to convect the connective tissue. The first term on the right hand side of Eq. (1) is the source or generation term, where g(c) = (1 - tanh[a₁(c -a₂)])/2.

NUMERICAL RESULTS

A simple explicit finite difference scheme with upwinding is used to solve the above set of PDEs numerically. We have used Dx = 0.05 and Dt = 0.001. See Ref. 13 for details.

Calculations were performed in with symmetric boundary condition at r=0 for both m(r,t) and c(r,t) . As r approaches infinity, both m(r,t) and c(r,t) should approach their initial values at infinity. In numerical calculations, we make sure that the spatial domain is large enough that the concentrations in a region at the rigit edge (r = r_max) of the spatial domain do not change from their initial values at the end of the simulations. Distribution of m(r,t) and c(r,t) for k =5 and b=2 at t=200 for the Cartesian, cylindrical and spherical geometires are shown in Fig. 1. In Cartesian geometry (la), the distribution at t = 200 is a sequence of thin regions (walls) of dense CT separated by tumor cells. Though the resulting structure resembles a nodular tumor, unfortunately-unlike in nodular tumors-in Cartesian coordinates, the innermost spike is the most dense, and the density at the first three spikes decreases with increasing distance of those spikes from the origin. However, the evolution in the cylindrical geometry (n = I) is somewhat different. The physical process is the same as in the Cartesian geometry, but the cylindrical geometry effect leads to unevenly spaced spikes with mixed amplitudes. Figure lb shows the m and c distribution as a function of r in cylindrical geometry at t = 200. In spherical geometry (n = 2), Fig. lc, the innermost spike has the smallest magnitude, mimicking the structure in a nodular tumor. Moreover the results with n = 2 also suggest that the lobes of tumorous cells are expected to be bigger near the center and relatively smaller near the capsule. Additional results are presented in the complete paper.

REFERENCES

1. Dvorak, H. F. 1986. Tumors: Wounds that do not Heal-Similarities Between Tumor Stroma Generation and Wound Healing. N. Engl. J. Med. 315: 1650-1659.
2. Barr, L. C., R. L. Carter & A. J. S..Davies. 1988. Encapsulation of Tumors as a Modified Wound Healing Response. Lancet. 135-137 , July 16, 1988.
3. Barr, L. C. 1980. The encapsulation of tumours [MS Thesis]. University of Manchester. Manchester, UK.
4. Adam, J. A. 1986. A Simplified Mathematical Model of Tumor Growth. Math. Biosci. 81: 229-244.
5. Adam, J. A. 1987. A Mathematical Model of Tumor Growth II: Effects of Geometry and Spatial Nonuniformity on Stability. Math. Biosci. 86: 183-211.
6. Adam, J. A. 1987. A Mathematical Model of Tumor Growth III: Comparison with Experiment. Math. Biosci. 86: 213-217.
7. Vaidya, V. G. & F. J. Alexandro, Jr. 1982. Evaluation of Some Mathematical Models for Tumor Growth. Int. J. Bio-Medical Computing. 13: 19-35.
8. Sachdev, P. L. 1987. Nonlinear Diffusive Waves. Cambridge University Press. Cambridge, UK.
9. Newman, W. I. 1980. Some Exact Solutions to a Non-linear Diffusion Problem in Population Genetics and Combustion. J. Theor. Biol. 85: 325-334.
10. Murray, J. D. 1989. Mathematical Biology. Springer-Verlag.
11. Rizwan-uddin. 1997. An Improved Coarse-Mesh Nodal Integral Method for Partial Differential Equations. Num. Methods for Partial Diff. Equations. 12: 113-145.
12. Sherrat, J. A. & M. A. Nowak. 1992. Oncogenes, Anti-oncogenes and the Immune Response to Cancer: A Mathematical Model. Proc. R. Soc. Lond., B, 248: 261-271.
13. Perumpanani, A.J., J.A. Sherratt & J. Norbury. 1996. Mathematical Modeling of Capsule Formation and Multi Nodularity in Benign Tumor Growth. Warwick Preprint: 51/1996, University of Warwick. Coventry, UK.

INTRODUCTION

The accumulation of cholesterol and other lipids within the intima is a preliminary stage in the atherogenic process.¹ These deposits are believed to be derived mainly from plasma lipoproteins, and particularly low-density lipoproteins (LDL). Experimental results² suggest that the flux of LDL into the arterial wall is not regulated by endothelial LDL receptors. Some cholesterol may seep into the arterial wall by infiltrating through leaky endothelial cell junctions. This lipid infiltration should depend on the concentration of lipids at the blood/vessel wall interface, as the blood vessel wall is directly exposed to the luminal surface lipid concentration. Because of regional differences in the near-wall blood flow velocity, blood pressure, and vascular permeability, the atherogenic lipid concentration at the luminal surface may vary according to location in the arterial tree. We therefore hypothesized that these local variations in the luminal surface lipid concentration may contribute to the localization of atherosclerosis. To substantiate the proposed hypothesis, we studied lipid transport from flowing blood to the arterial wall under both steady-state flow³ and pulsatile flow conditions.

METHOD

To calculate the luminal surface concentration of LDL, the following assumptions were made: (1) the fluid (blood) is homogeneous, incompressible, and Newtonian, with a constant viscosity of 0.035 gr/cm.sec and a density of 1.05 gr/cm³; (2) the blood vessel is a straight cylindrical tube with a uniform internal diameter; (3) the vessel wall is permeable to plasma and has a filtration rate of the order of 10^-6 cm/sec;⁴ and (4) the convective and diffusive flux of LDL into the vessel wall are so small that their effect on the luminal surface LDL concentration is negligible. The lipoprotein accumulation at the luminal surface of the artery was determined by the balance between convective and diffusive transport, and the no-slip and permeable wall conditions was applied at the artery wall. At the inlet, the LDL concentration was assumed to be uniform. The blood flow was reproduced by seventeen first harmonics of the flow waveform in the human carotid artery.⁵ Fully developed velocity profiles corresponding to the pulse waveform were calculated and used as the inlet boundary condition.

Under these conditions, the Navier-Stokes equations and a convective and diffusive mass transfer equation were solved numerically. The flow field equations were made discrete by the DuFort-Frankel Leapfrog method,6 and the Crank-Nicolson method, was used to solve the time-dependent mass transfer equation.

RESULTS

Numerical simulations were carried out on human common carotid arteries with an internal diameter of 7.0 mm and a time-averaged flow rate of 275 ml/min.⁵ The heart beat rate was taken as 70/min. Under these conditions, the time-averaged Reynolds number was 250 and the frequency parameter (Womersley number based on the fundamental frequency), a was 5.2, and the normalized angular frequency,w was 0.2. The LDL concentration was calculated for a filtration rate of 10^-6 to 10^-5 cm/sec and for a Schmidt number of 1.67 x 10⁵ to 6.67 x 10⁵, corresponding to a LDL diffusion coefficient of 5.0 x 10^-8 to 2.0 x 10^-7 cm²/sec.^7,8 The results revealed that under both steady-state and pulsatile flow conditions, low molecular diffusivities of LDL led to a concentration boundary layer at the vessel wall, in which the LDL concentration was apparently much higher than the bulk concentration. The effect of the LDL diffusion coefficient on the luminal surface LDL concentration, was studied by assuming that the filtration rate remained constant. FIGURE 1 shows the plot of relative surface concentration (steady periodic state values) against time in one cardiac cycle. As shown in this figure, in one cardiac cycle, there was a short period of time during which C_w /C₀ was lower than the value under steady-state flow. However, the time-average value was slightly higher than that of corresponding steady flow case. FIGURE 2a presents the effect of the filtration rate on the luminal surface LDL concentration. The time-average value of the dimensionless luminal surface LDL concentration increased linearly with increasing filtration rate. In an artery with a normal physiological filtration rate (4.0 x 10^-6 cm/sec), the time-average value of the luminal surface LDL concentration was 5 to 14% greater than that in the bulk flow. FIGURE 2b shows the plot of relative time-averaged luminal surface LDL concentration, C_w /C₀, against time-averaged wall shear rate (flow rate), . The relative time-averaged luminal surface LDL concentration decreased sharply at low wall shear rate and then approached a constant value asymptotically as wall shear rate was increased. Similar to the steady state case,³ the relative time-averaged luminal surface LDL concentration was more sensitive to changes in flow conditions at low flow rates.

CONCLUSION

"Concentration polarization" of lipids may occur at the blood/arterial wall interface under physiological pulsatile flow conditions. The luminal surface LDL concentration at the arterial wall is flow-dependent, and varies linearly with the filtration rate through the vessel wall and inversely with wall shear rate. This lipid transport phenomenon within the circulation may have important implications for the pathogenesis and localization of vascular disorders.

ACKNOWLEDGMENTS

The first author wishes to acknowledge the scholarship awarded by the Ministry of Culture and Higher Education of Iran. The technical assistance of Claire Kingston was appreciated.

REFERENCES

1. Stary, H.C. 1989. Evolution and progression of atherosclerotic lesions in coronary arteries of children and young adults. Arteriosclerosis 9 (Suppl I): I19-I32.
2. Wiklund, O., T.E. Carew & D. Steinberg. 1985. Role of the low density lipoprotein receptor in penetration of low density lipoprotein into rabbit aortic wall. Arteriosclerosis 5: 135-141.
3. Deng, X.Y., Y. Marois, T. How, Y. Merhi, W.M. King & R. Guidoin. 1995. Luminal surface concentration of lipoprotein (LDL) and its effect on the wall uptake by canine carotid arteries. J. Vasc. Surg. 21: 135-145.
4. WILENS, S.L. & R.T. MCLUSKEY. 1952. The comparative of filtration properties of excised arteries and veins. Am. J. Med. Sci. 224: 540-547.
5. Bharadvaj, B.K., R.F. Mabon & D.P. Giddens. 1982. Steady flow in a model of the human carotid bifurcation. Part I-flow visualization. J. Biomech. 15: 349-362.
6. Roache, P.J. 1972. Computational Fluid Dynamics. Hermosa Publisher, Albuquerque.
7. Caro, C.G., J.M. Fitz-Gerald & R.C. Schroter. 1971. Atheroma and arterial wall shear: Observation, correlation, and proposal for a shear dependent mass transfer mechanism for atherogenesis. Proc. Royal. Soc. Lond. B117: 109-159.
8. Back, L.H. 1975. Theoretical investigation of mass transport to arterial walls in various blood flow regions - I. Flow field and lipoprotein transport. Math. Biosci. 27: 231-262.

KEYWORDS

Textural Characteristics, Polarisation Microscopy, Dendrites, Genetic, Ozone, Aerosol, Oxygen, Biocolloid.

The free radical peroxidation of the lipids was carried out in vitro in the aerosol regime by passing through them ozone-oxygen mixture, which was obtained in a barrier discharge reactor from the 1,5 % ozone. Systematic investigations of the blood plasma from the donors (20 test) and patients with myocardial infarction (50 test) were carried out. Textural characteristics (1) were received in the flat capillaries of the "sandwich" type (2) and were revealed on the polarisation microscope MIN-8. The flat capillary had the square ~ 1 sm² and thickness ~ 0,01 nm. The inner surface of the flat capillary in the glow discharge reactor was covered by a thin polymer film with the surface tension coefficient (2), that had regulating polar and dispersion constituents. The value polar and dispersion constituents were defined by the solution of the equation system on a computer. The optical parameters of the blood plasma were registered in the range 500-4000 sm^-1 (on device Specord-80M) and 10-600 sm on Furrier spectrometer with computing complex. The phase transition temperature in the model cellular-vesicular structure (gel-liquid crystal state) was defined from dependence on conductivity within co-ordinates (logarithm of electrical conductivity of blood plasma - 1/T [Kelvin^-1]). The results of the model experiments in vitro were compared with the effects of the medicinal antioxidants in vivo.

It was stated, that when are changing the current in the reactor of glow discharge (0-10^-2 A), the processing time of capillary (0-5400 sec), the flow speed of the mixture gentle gas-monomer (methacryl acid and others) V=(2,5-5,0)*10^-3 H/m, under g_s^d:g_s^p=1-10. The surface energy was calculated as:

Craft points placement was defined by scan calometry method. Some results of the investigations of the lipids radical peroxide oxidation influence are shown on the figures 1,2.

With the removal of the water from the flat capillary, the dendrites, spherolytes, fan- and needle-shaped crystals in the solid phase were registered in the blood plasma at patients with myocardial infarction and the lines of liquid crystals at donors. It was indicated, that the maximal speed of crystallisation (up to 0.2 mm/sec) was on the layers with state g_s^d:g_s^p=10. The layers of this quality at the first time helped to receive the pseudo isotropic textures with size up to 10 mm of the cellular type in the lamella phase in the flat capillary. The textures of the regular pentagonal and hexagonal shape with 90^o (rotation of the polarisation plane of the membrane wall with meeting in the node under angle 120^o (are presenting especial interest. In our opinion, it is connected with rotational isomerisation (rot[+] and conformation rot[-]) as well, as with the weak interaction of non-polar biomolecular groups and their location in the dodecahedral regions in the combination with the cavities of 14-hedrons, forming clatter - similar solution. The textural research complies with the results of the biochemical and spectral analysis. The Furrier-spectroscopy helped to specify the stratification of the biocolloid and the breaking hydrogen connections during dismethabolism. The metabolic disorder in case of myocardial infarction, the simulation of peroxide oxidation of the lipids by the ozone treatment of the blood plasma in vitro and the action of the antioxidants in vivo are fixed in the infrared spectrum by changing the line intensifies of non-saturated connections of the arachidonic and linolenic acids -C=C-C=C- (l~1680-1580; 1070 sm^-1) and catabolism products groups =COC= (l~3050-2990 sm^-1), -C=C- (l~2140 sm^-1). The analysis of the lines -P-O-C₆H₅ (l~1240-1990 sm^{- 1}) and -P-O-R (l~196001940 sm^-1) is promising. The changing of the spectra intensity and breaking the regular textures during ozone treatment with the rising of total area of the optically active groups, allow to make a conclusion, that the ability of a blood plasma to the lyomesomorphism and structuring indicates the infringement of recyclation of the membrane material and of metabolism.

Figure 1 The influence of free radical oxidation of lipids on the Craft points placement:

1. Donor's blood plasma;
2. Blood plasma + 120 relative points of O3

The human blood plasma proteins realise creating bonds mean the information transfer, influencing on the genetic mechanism of the cell. Every aminoacids are according from 1 (Met, Trp) to 6 (Leu, Arg, Ser) of the codones (the code which has the abundance properties) (the table 1). It was found that the needle crystal textures of the blood plasma of the patients with myocardial infarction, in the case closer to death are characteristic of the aminoacids textures too. The high sensitivity of the polarising microscopy method let to recommend the methods as a one from the long- term other methods of the genetic breaches find.

Table 1. The aminoacids, its conventional signs, (three and one letter symbol) and the codones are according to them, the crystal type.

The investigations of the temperature dependence of the conductivity (lgs-1/T) at the different degrees of the plasma ozone treatments in vitro helped to show for the first time the quantitative influence of the lipids peroxide oxidation on the conductivity [lgs=(-10.5)-(-9.7)] as well, as more abrupt phase transition in the temperature range [1/T=3.15-3.09 K^-1].

ACKNOWLEDGEMENTS

We would like to thank N. Kaledenkova for her help to carry out the experiment and A.Kochetov and A.Wyrleiev for their help to put into the form this paper.

REFERENCE

1. Zaitsev V.V., Zaitseva N.B., Usol'tseva N.V. Orientation of Lyotropic Liquid Crystals of Blood Plasma // Colloid Journal (Russia), 1996; 58; 713-716.
2. Zaitsev V.V., Zaitseva N.B., Usol'tseva N.V. The Textures of the Biological Liquid Crystals of a Patients with Myocardial Infarction // Academy of Science News (Russia), 1996; 60; 115-118.