James W. Baish

Fractal Characteristics of Tumor Vascular Architecture During Tumor Growth and Regression

Objective: Tumor vascular networks are different from normal vascular networks, but the mechanisms underlying these differences are not known. Understanding these mechanisms may be the key to improving the efficacy of treatment of solid tumors.

Heat Transport Mechanisms in Vascular Tissues: A Model Comparison

We have conducted a parametric comparison of three different vascular models for describing heat transport in tissue. Analytical and numerical methods were used to predict the gross temperature distribution throughout the tissue and the small-scale temperature gradients associated with thermally significant blood vessels. The models are: an array of unidirectional vessels, an array of countercurrent vessels, and a set of large vessels feeding small vessels which then drain into large vessels. We show that three continuum formulations of bioheat transfer (directed perfusion, effective conductivity, and a temperature-dependent heat sink) are limiting cases of the vascular models with respect to the thermal equilibration length of the vessels. When this length is comparable to the width of the heated region of tissue, the local temperature changes near the vessels can be comparable to the gross temperature elevation. These results are important to the use of thermal techniques used to measure the blood perfusion rate and in the treatment of cancer with local hyperthermia.

Scaling rules for diffusive drug delivery in tumor and normal tissues

Delivery of blood-borne molecules and nanoparticles from the vasculature to cells in the tissue differs dramatically between tumor and normal tissues due to differences in their vascular architectures. Here we show that two simple measures of vascular geometry—δmax and λ—readily obtained from vascular images, capture these differences and link vascular structure to delivery in both tissue types. The longest time needed to bring materials to their destination scales with the square of δmax, the maximum distance in the tissue from the nearest blood vessel, whereas λ, a measure of the shape of the spaces between vessels, determines the rate of delivery for shorter times. Our results are useful for evaluating how new therapeutic agents that inhibit or stimulate vascular growth alter the functional efficiency of the vasculature and more broadly for analysis of diffusion in irregularly shaped domains.

Cancer, angiogenesis and fractals.

To the editor—The recent commentary by Coffey emphasizes the value of nonlinear dynamics and fractals as tools for studying aspects of biological pattern formation that are not readily accessible to strictly molecular methods. Our studies of the vascular architecture in tumors support Coffey’s points about the delicate interplay among gradients in diffusible cytokines, heterogeneity of cell pheno-

Small-scale temperature fluctuations in perfused tissue during local hyperthermia

We develop analytical expressions (scaling laws) for the local temperature fluctuations near isolated and countercurrent blood vessels during hyperthermia. These scaling laws relate the magnitude of such fluctuations to the size of the heated region and to the thermal equilibration length of the vessels. A new equilibration length is identified for countercurrent vessels. Significant temperature differences are predicted between the vessels and the immediately adjacent tissue when the equilibration length is comparable to or longer than the size of the heated tissue region. Countercurrent vessels are shown to have shorter equilibration lengths and produce smaller temperature fluctuations than isolated vessels of the same size.

Solid stress and elastic energy as measures of tumour mechanopathology

Solid stress and tissue stiffness affect tumour growth, invasion, metastasis and treatment. Unlike stiffness, which can be precisely mapped in tumours, the measurement of solid stresses is challenging. Here, we show that two-dimensional spatial mappings of solid stress and the resulting elastic energy in excised or in situ tumours with arbitrary shapes and wide size ranges can be obtained via three distinct and quantitative techniques that rely on the measurement of tissue displacement after disruption of the confining structures. Application of these methods in models of primary tumours and metastasis revealed that: (i) solid stress depends on both cancer cells and their microenvironment; (ii) solid stress increases with tumour size; and (iii) mechanical confinement by the surrounding tissue significantly contributes to intratumoural solid stress. Further study of the genesis and consequences of solid stress, facilitated by the engineering principles presented here, may lead to significant discoveries and new therapies.

Viscous damping of vibrations in microtubules.

Pokorný et al. have recently suggested that metabolic processes drivemicrotubules in a cell to vibrate at Megahertz frequencies, but the theorydoes not explicitly consider dissipative effects which will tend to damp outthe vibrations. To examine the effects of viscous damping on the structure,we determine viscous forces and rate of energy loss in a cylinderundergoing longitudinal oscillations in water. A nondimensional expressionis obtained for the viscous drag on the cylinder. When applied to amicrotubule, the results indicate that viscous damping is several orders ofmagnitude too large to allow resonant vibrations.

Mechanobiological oscillators control lymph flow

Significance Dysfunction of lymphatic drainage results in significant morbidity in millions of patients each year, and there are currently no pharmacological treatments available. Because a mechanistic understanding of lymphatic control has been elusive, it has not been possible to develop effective, targeted therapies to alleviate lymphedema or facilitate immune cell trafficking in lymphatic pathologies. Here, we show that complementary biological oscillators control lymphatic transport, driven by mechanosensitive calcium and nitric oxide (NO) dynamics. The mechanism shows fascinating adaptability and autoregulation, inducing active pumping of the lymphatic vessels when gravity opposes flow, but vessel relaxation when pressures are able to drive flow passively. The ability of cells to sense and respond to physical forces has been recognized for decades, but researchers are only beginning to appreciate the fundamental importance of mechanical signals in biology. At the larger scale, there has been increased interest in the collective organization of cells and their ability to produce complex, “emergent” behaviors. Often, these complex behaviors result in tissue-level control mechanisms that manifest as biological oscillators, such as observed in fireflies, heartbeats, and circadian rhythms. In many cases, these complex, collective behaviors are controlled—at least in part—by physical forces imposed on the tissue or created by the cells. Here, we use mathematical simulations to show that two complementary mechanobiological oscillators are sufficient to control fluid transport in the lymphatic system: Ca2+-mediated contractions can be triggered by vessel stretch, whereas nitric oxide produced in response to the resulting fluid shear stress causes the lymphatic vessel to relax locally. Our model predicts that the Ca2+ and NO levels alternate spatiotemporally, establishing complementary feedback loops, and that the resulting phasic contractions drive lymph flow. We show that this mechanism is self-regulating and robust over a range of fluid pressure environments, allowing the lymphatic vessels to provide pumping when needed but remain open when flow can be driven by tissue pressure or gravity. Our simulations accurately reproduce the responses to pressure challenges and signaling pathway manipulations observed experimentally, providing an integrated conceptual framework for lymphatic function.

Towards Optimal Design of Cancer Nanomedicines: Multi-stage Nanoparticles for the Treatment of Solid Tumors

Conventional drug delivery systems for solid tumors are composed of a nano-carrier that releases its therapeutic load. These two-stage nanoparticles utilize the enhanced permeability and retention (EPR) effect to enable preferential delivery to tumor tissue. However, the size-dependency of the EPR, the limited penetration of nanoparticles into the tumor as well as the rapid binding of the particles or the released cytotoxic agents to cancer cells and stromal components inhibit the uniform distribution of the drug and the efficacy of the treatment. Here, we employ mathematical modeling to study the effect of particle size, drug release rate and binding affinity on the distribution and efficacy of nanoparticles to derive optimal design rules. Furthermore, we introduce a new multi-stage delivery system. The system consists of a 20-nm primary nanoparticle, which releases 5-nm secondary particles, which in turn release the chemotherapeutic drug. We found that tuning the drug release kinetics and binding affinities leads to improved delivery of the drug. Our results also indicate that multi-stage nanoparticles are superior over two-stage nano-carriers provided they have a faster drug release rate and for high binding affinity drugs. Furthermore, our results suggest that smaller nanoparticles achieve better treatment outcome.

Role of vascular normalization in benefit from metronomic chemotherapy

Significance Metronomic dosing has been proposed as an alternative to maximum tolerated doses of chemotherapy. Even though this strategy has shown promise in improving therapeutic outcomes in some cases, the underlying mechanisms of action are not fully understood. In this study, using mathematical modeling, we show that metronomic chemotherapy could be beneficial by normalizing tumor blood vessel function and improving tumor perfusion. Improved perfusion enhances the delivery of drugs to solid tumors and alleviates hypoxia. These effects also result in an improved immune response, contributing to increased killing of cancer cells, including more resistant cancer stem-like cells. Metronomic dosing of chemotherapy—defined as frequent administration at lower doses—has been shown to be more efficacious than maximum tolerated dose treatment in preclinical studies, and is currently being tested in the clinic. Although multiple mechanisms of benefit from metronomic chemotherapy have been proposed, how these mechanisms are related to one another and which one is dominant for a given tumor–drug combination is not known. To this end, we have developed a mathematical model that incorporates various proposed mechanisms, and report here that improved function of tumor vessels is a key determinant of benefit from metronomic chemotherapy. In our analysis, we used multiple dosage schedules and incorporated interactions among cancer cells, stem-like cancer cells, immune cells, and the tumor vasculature. We found that metronomic chemotherapy induces functional normalization of tumor blood vessels, resulting in improved tumor perfusion. Improved perfusion alleviates hypoxia, which reprograms the immunosuppressive tumor microenvironment toward immunostimulation and improves drug delivery and therapeutic outcomes. Indeed, in our model, improved vessel function enhanced the delivery of oxygen and drugs, increased the number of effector immune cells, and decreased the number of regulatory T cells, which in turn killed a larger number of cancer cells, including cancer stem-like cells. Vessel function was further improved owing to decompression of intratumoral vessels as a result of increased killing of cancer cells, setting up a positive feedback loop. Our model enables evaluation of the relative importance of these mechanisms, and suggests guidelines for the optimal use of metronomic therapy.