By: Zarna.A.Bhavsar 11/25/2008

Transcription

1 Transport of Molecules, Particles, and Cells in Solid Tumors A Model for Temporal heterogeneities of tumor blood flow By: Zarna.A.Bhavsar 11/25/2008 Contents Background Approaches Specific aims Developments because of tumor Angiogenesis Blood flow heterogeneities Metabolic microenvironment Difficulty in the transport of molecules, particles and cells Transport across the microvascular walls Transport through the interstitial space Transport of cells Pharmacokinetic modelling Bench to bedside Mathematical modeling Results and discussion Limitation Conclusion

2 Cancer Type of deadly disease in which a group of cells display uncontrolled growth, invasion and sometimes metastasis along with the formation of tumors In U.S and other developed countries, around 25% of the total deaths occur because of cancer Advances in medical science have led to the development of therapeutic anti cancer drugs The blood borne particles, molecules or cells have to pass through this abnormal vascular compartment, transport through the microvascular wall and traverse through the interstitial compartment in order to reach the cancer cells.

3 Approaches A tissue isolated tumor connected by a single artery and a single vein Angiogenesis assay was done on Sandison rabbit ear chamber, a modified Algire mouse dorsal chamber and a cranial window in mice and rats to study the physiology of vessels induced by individual growth factors To assess the deformability, adhesion, permeability and growth stress of normal and neoplastic cells by using in vitro tests Various molecular biology techniques and green fluorescence protein are used to monitor the promoter activities Mathematical methods are proposed to describe and integrate the data obtained from the above approaches and to scale up the biodistribution data from mice to men Specific aims Understand how angiogenesis takes place and what determines blood flow heterogeneities in tumors How blood flow influences the metabolic microenvironment in tumors and how microenvironment affects the biological properties in tumor How material moves across the microvascular walls How material moves through the interstitial compartment and the lymphatics The role of cell deformation and adhesion in the delivery of cells The scale up from mice to men To propose a mathematical model for temporal heterogeneties of tumor blood flow

4 Angiogenesis Development of new blood vessels Necessary for the growth of tumor, metastasis, supply of oxygen and nutrients to the tumor Tumor vessels grow by various mechanisms: (1)The host vascular network expands by budding of endothelial sprouts or formation of bridges (angiogenesis) (2) tumor vessels remodel and expand by the insertion of interstitial tissue columns into the lumen of pre existing vessels (intussusception) (3) endothelial cell precursors (angioblasts) home from the bone marrow or peripheral blood into tumors and contribute to the endothelial lining of tumour vessels (vasculogenesis). Lymphatic vessels around tumors drain the interstitial fluid and provide a gateway for metastasizing tumor cells.

5 Blood Flow Heterogeneities Tumor blood vessels arise from the host vasculature but still they have completely different organization depending on the tumor type, its growth rate and its location. Blood flow heterogeneties arise because of the following reasons: Elevated geometric and viscous resistence in tumor vessels Coupling between high vascular permeability and elevated interstitial fluid pressure Vascular remodeling by intussusception Solid stress generated by proliferating cancer cells Blood flow heterogeneities result in a metabolic microenvironment: Decrease in ph and po2 with increase in distance from the tumor Hypoxic region in the center of the tumor problem in the interstitial delivery of drug Transport across the microvascular wall Extravasation on reaching the exchange vessel dependent on the phenomenon of diffusion, convection and to some extent on transcytosis The transport parameters that govern the transport of a molecule across normal or tumor vessels are the surface area of exchange, transvascular concentration and the pressure gradients Although the tumor vessels are leaky, the extravasation of molecules is poor in various regions of the tumor due to the following reason: tumors exhibit high interstitial fluid pressure(ifp) high pressure drops to the normal value in the periphery of the tumor oncotic and hydrostatic pressures are equal between the intravascular and extravascular space so extravasation is lowered in high pressure regions The average vascular surface area per unit of tissue weight decreases with the tumor growth so transvascular exchange is less in large tumors compared to small tumors

6 Transport through interstitial space On extravasation the transport of molecule through the interstitial space is governed by the diffusion and convection mechanisms Diffusion: By Fick s law diffusion is proportional to the concentration gradient J= D ф/ x Convection: Similarly, the convective flux, Jc given by: Jc = CRFu = CRFK ( p/ x) u=convective flow velocity of the solvent resulting from pressure gradients in the medium RF = the retardation factor (solute convective velocity/solvent convective velocity) K =the tissue hydraulic conductivity for convective flow of solvent through the medium (k/ή,where k is Darcy's constant, and ή is solvent viscosity) and p/ x is the pressure gradient (p hydrostatic pressure) Even though the values of D (diffusion coefficient) and K (interstitial hydraulic conductivity) are high the macromolecules injected in the tumor take much time to diffuse The time constant for a molecule with diffusion coefficient D to diffuse across distance L is given by t=(l^2/4d) As the distance increases the time for the diffusion also increases There would be no delivery of macromolecules by the vessels if the central vessels are completely collapsed because of cellular proliferation and interstitial matrix rearrangement

7 Transport of cells A leukocyte that enters the blood vessel may continue to move along with the flowing blood, collide with the vessel wall, adhere transiently or stably and finally extravasate These actions are governed by: (1) Local hydrodynamic forces determined by vessel diameter and fluid velocity (2) Adhesive forces expression, strength, kinetics of bond formation between adhesion molecules and surface area of contact Pharmacokinetic modeling: Scale up from mouse to human Scale up the biodistribution of low molecular weight agents and extending it to macromolecules and cells Parameters like volume and blood flow rate for the vascular, interstitial and cellular regions which are not known are estimated by using the known parameters and fitting the model to the murine biodistribution data Scaling up the parameters by using the well defined scale up laws the biodistribution in human patients can be predicted and compared with clinical data Differences between predictions and actual data are useful in identifying differences between different species

8 Bench to bedside The interstitial fluid pressure(ifp) rises steeply in the tumor boundary needle is designed with a pressure sensor on the needle needle localizes the tumor for surgical excision helps localize early disease increase the interstitial transport rate of molecules by increasing K or D enzymatically A MODEL FOR TEMPORAL HETEROGENEITIES FOR TUMOR BLOOD FLOW TBF exhibits spatial and temporal heterogeneities which affect tumor growth, metastases and therapy Spatial heterogeneity: the tumor blood flow varies according to the distance from the tumor the blood is diverted away from the center of the tumor towards the peripheral path which leaves a scarcely perfused area in the middle of the tumor the central part of the tumor being hypoxic, the drug delivery in this area is ineffective Temporal heterogeneity: the tumor blood flow varies according to the changes in time Even at a fixed location, the blood flow is not uniform with time and can exhibit intermittent flow rate accompanied by inversions in the direction of flow Related to the phenomenon of intussusception

9 Mathematical modeling for temporal inhomogeneities A tumor capillary of length L and thickness having a uniform cross section with a constant width w and a variable height 2h is modeled as a two dimensional channel Fig. 2. A representative tumor capillary of length L is modeled as indicated; x is the axial coordinate, h(x, t) is half of the open gap of the capillary. The deformation is assumed to be symmetric with respect to the x axis. Buckling mechanism The central part of the vessel changes its height but the width remains unchanged when the vessel is completely collapsed there are two small apertures at the vessel sides which allow some amount of fluid to flow through The cross section of a collapsing capillary under an applied external pressure, which is greater than the internal pressure. Closing occurs through a buckling mechanism.

10 Mathematical Model Equations The net pressure acting on the section of capillary located at x at time t is given by: π(x,t) is pressure varying from arterial to venous pressure and πi is the uniform and constant IFP(Interstitial Fluid Pressure). Assuming the thickness δ to be constant and small enough,the deformation of the capillary can be given by: h(x,t) height varying from arterial to venous pressure and h0 is the initial height of the capillary. where T is the constant tension in the capillary walls, c is drag coefficient, ρ is the density of interstitial fluid and Habs is the virtual mass coefficient. The last two terms account for the drag effect on vessel membrane and socalled virtual mass effect Mathematical Model Equations The virtual mass effect occurs when a solid accelerates within a fluid at rest. The function φ is the capillary stiffness function given by: where E is the Young Modulus of the capillary and K is the bending stiffness This relationship is often referred to as tube law and discriminates the cases in which the capillary is being inflated by the internal blood flow or buckles under the external IFP. Assuming 1 D flow and Newtonian fluid, neglecting separation effects the flow rate is given by: The equation of conservation of mass for the blood flow is: where Vp and Vw denote the fluid perfusing through porous wall and capillary wall.

11 Boundary Conditions for Mathematical Model The proper initial and boundary conditions for the present problem are: Results Fig.4. The shape of the deformed capillary at the initial time and at two subsequent instants in the upper figure and corresponding MVP distribution in the lower figure.(the arrow is indication of increasing time)

12 Fig.5. The plot of arterial (Qa) and venous (Qv) flow rate vs time. The behavior is oscillatory with irregular amplitude. Discussion Capillary shows diffused stenosis towards the section exit. Pressure increases as capillary collapses. Increase in pressure works against IFP, reopens capillary and hence leading to decrease in pressure. From Fig.5. the exit flow rate Qv is greater than entrance flow rate Qa due to negative transmural pressure. (Not true for tumor due to fluid leakage) Tumor does not show equilibration between average MVP inside leaky tumor and IFP outside vessel. Hence tumor flow is driven mainly due to difference between arterial and venous pressure. Closeness of average MVP and IFP also leads to unsteady behavior of tumor blood flow.

13 Disadvantages of study The oscillations have a frequency much higher than the frequency observed in vivo. The model does not predict the periodic inversion of the direction of flow. Considered only a straight blood vessel while in reality a more complicated vessel network. Vessel growth was also neglected in the model. Conclusion Understanding of the transport mechanism of the drug particles through the tumor vessels and the tumor cells can be very useful in developing potential drug delivery systems for delivering the therapeutic anticancer agents to kill the tumor cells for the treatment of cancer

14 References 1. Rakesh K. Jain, Transport of Molecules, Particles and Cells in Solid Tumors 2. Francesco Mollica, Rakesh.K.Jain, and Paolo A. Netti, A model for temporal heterogeneities of tumor blood flow 3. Matthew R. Dreher, Wenge Liu, Charles R. Michelich, Mark W. Dewhirst, Fan Yuan, Ashutosh Chilkoti Tumor Vascular Permeability, Accumulation, and Penetration of Macromolecular Drug Carriers 4. Rakesh K. Jain, et al. 2 Science 307, 58 (2005); Normalization of Tumor Vasculature: An Emerging Concept in Antiangiogenic Therapyrging 5. Peter Carmeliet* & Rakesh K. Jain Angiogenesis in cancer and other diseases 6. Carolyn Vachani, MSN, RN, AOCN Understanding Your Pathology Report: Colon Cancer 7. Maria Grazia Sacco, Enrica Mira Catò, Francesca Faggioli Institute for biomedical technologies national research council Thank you!

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