A Century of Fractionated Radiotherapy: How Mathematical Oncology Can Break the Rules.

Radiotherapy is involved in 50% of all cancer treatments and 40% of cancer cures. Most of these treatments are delivered in fractions of equal doses of radiation (Fractional Equivalent Dosing (FED)) in days to weeks. This treatment paradigm has remained unchanged in the past century and does not account for the development of radioresistance during treatment. Even if under-optimized, deviating from a century of successful therapy delivered in FED can be difficult. One way of exploring the infinite space of fraction size and scheduling to identify optimal fractionation schedules is through mathematical oncology simulations that allow for in silico evaluation. This review article explores the evidence that current fractionation promotes the development of radioresistance, summarizes mathematical solutions to account for radioresistance, both in the curative and non-curative setting, and reviews current clinical data investigating non-FED fractionated radiotherapy.

Clinically validated model predicts the effect of intratumoral heterogeneity on overall survival for non-small cell lung cancer (NSCLC) patients.

BACKGROUND AND OBJECTIVE: Radiation therapy is used in nearly 50% of cancer treatments in the developed world. Currently, radiation treatments are homogenous and fail to take into consideration intratumoral heterogeneity. We demonstrate the importance of considering intratumoral heterogeneity and the development of resistance during fractionated radiotherapy when the same dose of radiation is delivered for all fractions (Fractional Equivalent Dosing FED). METHODS: A mathematical model was developed with the following parameters: a starting population of 1011 non-small cell lung cancer (NSCLC) tumor cells, 48 h doubling time, and cell death per the linear-quadratic (LQ) model with α and ß values derived from RSIα/ß, in a previously described gene expression based model that estimates α and ß. To incorporate both inter- and intratumor radiation sensitivity, RSIα/ß output for each patient sample is assumed to represent an average value in a gamma distribution with the bounds set to -50% and +50% of RSIα/b. Therefore, we assume that within a given tumor there are subpopulations that have varying radiation sensitivity parameters that are distinct from other tumor samples with a different mean RSIα/ß. A simulation cohort (SC) comprised of 100 lung cancer patients with available RSIα/ß (patient specific α and ß values) was used to investigate 60 Gy in 30 fractions with fractionally equivalent dosing (FED). A separate validation cohort (VC) of 57 lung cancer patients treated with radiation with available local control (LC), overall survival (OS), and tumor gene expression was used to clinically validate the model. Cox regression was used to test for significance to predict clinical outcomes as a continuous variable in multivariate analysis (MVA). Finally, the VC was used to compare FED schedules with various altered fractionation schema utilizing a Kruskal-Wallis test. This was examined using the end points of end of treatment log cell count (LCC) and by a parameter described as mean log kill efficiency (LKE) defined as: LCC  = â€…log10(tumorcellcount) [Formula: see text] RESULTS: Cox regression analysis on LCC for the VC demonstrates that, after incorporation of intratumoral heterogeneity, LCC has a linear correlation with local control (p = 0.002) and overall survival (p = < 0.001). Other suggested treatment schedules labeled as High Intensity Treatment (HIT) with a total 60 Gy delivered over 6 weeks have a lower mean LCC and an increased LKE compared to standard of care 60 Gy delivered in FED in the VC. CONCLUSION: We find that LCC is a clinically relevant metric that is correlated with local control and overall survival in NSCLC. We conclude that 60 Gy delivered over 6 weeks with altered HIT fractionation leads to an enhancement in tumor control compared to FED when intratumoral heterogeneity is considered.

Bimolecular Master Equations for a Single and Multiple Potential Wells with Analytic Solutions.

The analytic solutions, that is, populations, are derived for the K-adiabatic and K-active bimolecular master equations, separately, for a single and multiple potential wells and reaction channels, where K is the component of the total angular momentum J along the axis of least moment of inertia of the recombination products at a given energy E. The analytic approach provides the functional dependence of the population of molecules on its K-active or K-adiabatic dissociation, association rate constants and the intermolecular energy transfer, where the approach may complement the usual numerical approaches for reactions of interest. Our previous work, Part I, considered the solutions for a single potential well, whereby an assumption utilized there is presently obviated in the derivation of the exact solutions and farther discussed. At the high-pressure limit, the K-adiabatic and K-active bimolecular master equations may each reduce, respectively, to the K-adiabatic and K-active bimolecular Rice-Ramsperger-Kassel-Marcus theory (high-pressure limit expressions) for bimolecular recombination rate constant, for a single potential well, and augmented by isomerization terms when multiple potential wells are present. In the low-pressure limit, the expression for population above the dissociation limit, associated with a single potential well, becomes equivalent to the usual presumed detailed balance between the association and dissociation rate constants, where the multiple well case is also considered. When the collision frequency of energy transfer, ZLJ, between the chemical intermediate and bath gas is sufficiently less than the dissociation rate constant kd( E' J' K') for postcollision ( E' J' K), then the solution for population, g( EJK)+, above the critical energy further simplifies such that depending on ZLJ, the dissociation and association rate constant kr( EJK), as g( EJK)+ = kr( EJK)A·BC/[ ZLJ+ kd( EJK)], where A and BC are the reactants, for example, relevant for O3 formation from O + O2 + Ar up to ∼100 bar; otherwise, additional contributions from postcollision are present and especially relevant at high pressures. In the aforementioned regime ZLJ < kd( E' J' K) the physical connection of recombination rate constants, krec based on either utilizing population from the master equation approach or a collision based bimolecular RRKM theory is traced and elucidated analytically that the rate constants are equal. Recombination rate constants, krec, based on the population, are also given and considered for an adiabatic or active K. For example, for O3 formation in Ar bath gas, the K-adiabatic-based krec for O3 yields 4.0 × 10-34 cm6 molecule-2 s-1 at T = 300 K and 1 bar, in agreement with the experimental value, where the contribution from the population of metastable ozone is discussed and the adiabaticity of K highlighted. A facile numerical implementation of the formalism for g( EJK) and krec for O3 is noted. The application of the expressions to ozone recombination as a function of pressure and temperature is given elsewhere, beyond the selection considered here, for studying the physical features, including the contributions from the K and intermolecular energy transfer to the krec, and the puzzles reported from experiments for this reaction.

Dynamics of 3D carcinoma cell invasion into aligned collagen.

Carcinoma cells frequently expand and invade from a confined lesion, or multicellular clusters, into and through the stroma on the path to metastasis, often with an efficiency dictated by the architecture and composition of the microenvironment. Specifically, in desmoplastic carcinomas such as those of the breast, aligned collagen tracks provide contact guidance cues for directed cancer cell invasion. Yet, the evolving dynamics of this process of invasion remains poorly understood, in part due to difficulties in continuously capturing both spatial and temporal heterogeneity and progression to invasion in experimental systems. Therefore, to study the local invasion process from cell dense clusters into aligned collagen architectures found in solid tumors, we developed a novel engineered 3D invasion platform that integrates an aligned collagen matrix with a cell dense tumor-like plug. Using multiphoton microscopy and quantitative analysis of cell motility, we track the invasion of cancer cells from cell-dense bulk clusters into the pre-aligned 3D matrix, and define the temporal evolution of the advancing invasion fronts over several days. This enables us to identify and probe cell dynamics in key regions of interest: behind, at, and beyond the edge of the invading lesion at distinct time points. Analysis of single cell migration identifies significant spatial heterogeneity in migration behavior between cells in the highly cell-dense region behind the leading edge of the invasion front and cells at and beyond the leading edge. Moreover, temporal variations in motility and directionality are also observed between cells within the cell-dense tumor-like plug and the leading invasive edge as its boundary extends into the anisotropic collagen over time. Furthermore, experimental results combined with mathematical modeling demonstrate that in addition to contact guidance, physical crowding of cells is a key regulating factor orchestrating variability in single cell migration during invasion into anisotropic ECM. Thus, our novel platform enables us to capture spatio-temporal dynamics of cell behavior behind, at, and beyond the invasive front and reveals heterogeneous, local interactions that lead to the emergence and maintenance of the advancing front.

Bimolecular recombination reactions: K-adiabatic and K-active forms of the bimolecular master equations and analytic solutions.

Expressions for a K-adiabatic master equation for a bimolecular recombination rate constant krec are derived for a bimolecular reaction forming a complex with a single well or complexes with multiple well, where K is the component of the total angular momentum along the axis of least moment of inertia of the recombination product. The K-active master equation is also considered. The exact analytic solutions, i.e., the K-adiabatic and K-active steady-state population distribution function of reactive complexes, g(EJK) and g(EJ), respectively, are derived for the K-adiabatic and K-active master equation cases using properties of inhomogeneous integral equations (Fredholm type). The solutions accommodate arbitrary intermolecular energy transfer models, e.g., the single exponential, double exponential, Gaussian, step-ladder, and near-singularity models. At the high pressure limit, the krec for both the K-adiabatic and K-active master equations reduce, respectively, to the K-adiabatic and K-active bimolecular Rice-Ramsperger-Kassel-Marcus theory (high pressure limit expressions). Ozone and its formation from O + O2 are known to exhibit an adiabatic K. The ratio of the K-adiabatic to the K-active recombination rate constants for ozone formation at the high pressure limit is calculated to be ∼0.9 at 300 K. Results on the temperature and pressure dependence of the recombination rate constants and populations of O3 will be presented elsewhere.

Bimolecular recombination reactions: K-adiabatic and K-active forms of RRKM theory, nonstatistical aspects, low-pressure rates, and time-dependent survival probabilities with application to ozone. 2.

We consider for bimolecular recombination reactions the K-adiabatic versus the K-active forms of RRKM theory, where K is the component of the total angular momentum along the axis of least moment of inertia of the recombination product. When that product is approximately a prolate symmetric top, with two moments of inertia of the product substantially larger than the third, K becomes a dynamically slowly varying quantity and the K-adiabatic form of RRKM theory is the appropriate version to use. Using classical trajectory results for the rate constant for ozone formation in the low-pressure region as an example, excellent agreement for the recombination rate constant k(rec) with the K-adiabatic RRKM theory is observed. Use of a two transition state (inner, outer TS) formalism also obviates any need for assessing recrossings in the exit channel. In contrast, the K-active form of RRKM theory for this system disagrees with the trajectory results by a factor of about 2.5. In this study we also consider the distribution of the (E, J) resolved time-dependent survival probabilities P(E, J, t) of the intermediate O3* formed from O + O2. It is calculated using classical trajectories. The initial conditions for classical trajectories were selected using action-angle variables and a total J representation for (E, J) resolved systems, as described in Part I.1 The difference between K-active and K-adiabatic treatments is reflected also in a difference of the K-active RRKM survival probability P(E, J, t) from its trajectory-based value and from its often non-single-exponential decay. It is shown analytically that krec (K-active) ≥ k(rec) (K-adiabatic), independent of the details of the TS (e.g., variational or fixed RRKM theory, 1-TS or 2-TS). Nonstatistical effects for O3* formation include a small initial recrossing of the transition state, a slow (several picoseconds) equipartitioning of energy among the two O-O bonds of the newly formed O3*, and a small nondissociation (a quasi-periodicity) of some trajectories originating in O3* (∼ 10%) and so, by microscopic reversibility, are not accessible from O + O2. An apparently new feature of the present results is the comparison of classical trajectories with K-adiabatic and K-active theories for rate constants of bimolecular recombinations. The quantum mechanical counterpart of classical K-adiabatic RRKM theory is also given, and its comparison with the experimental k(rec) for O3 is given elsewhere.

1H[19F] NOE NMR structural signatures of the insulin R6 hexamer: evidence of a capped HisB10 site in aryl- and arylacryloyl-carboxylate complexes.

NEW AND IMPROVED INSULIN: 1H[19F] NOE NMR difference spectra for CF(3)-substituted aromatic carboxylates bound at the HisB10 sites of the R(6) human insulin (HI) hexamer show strong NOEs between the CF(3) groups and the LeuB6, AsnB3, and PheB1 sidechains. The NOEs and structural modeling establish that these carboxylates form closed complexes with the HisB10 site capped by the PheB1 rings.

Insulin allosteric behavior: detection, identification, and quantification of allosteric states via 19F NMR.

The insulin hexamer is an allosteric protein widely used in formulations for the treatment of diabetes. The hexamer exhibits positive and negative cooperativity and apparent half-site binding activity, reflecting the interconversion of three allosteric states, designated as T6, T3R3, and R6. The hexamer contains two symmetry-related Zn2+ located 16 A apart on the 3-fold symmetry axis. In the transition of T3 units to R3 units, Zn2+ switches from an octahedral Zn2+ N3O3 complex (N is HisB10, O is H2O) to a distorted tetrahedral Zn2+ N3L complex (L is a monovalent anion). Hence, monovalent anions are allosteric ligands that stabilize R3 units of T3R3 and R6. Herein, we exploit the high sensitivity of 19F NMR chemical shifts and fluorinated carboxylates to reveal subtle differences in the anion-binding sites of T3R3 and R6. We show that the chemical shifts of 4- and 3-trifluoromethylbenzoate and 4- and 2-trifluoromethylcinnamate give bound resonances that distinguish between T3R3 and R6. 3-Trifluoromethylbenzoate and 2-trifluoromethylcinnamate also were shown to bind to the R3 units of T3R3 and R6 in two alternative, slowly interconverting modes with different microenvironments for the CF3 groups. Line width analysis shows that ligand off rates are slower by 1/10(3) than the diffusion limit, indicating a rate-limiting protein conformational transition. These studies confirm that the Seydoux, Malhotra, and Bernhard allosteric model (Bloom, C. R., Choi, W. E., Brzovic, P. S., Ha, J. J., Huang, S. T., Kaarsholm, N. C., and Dunn, M. F. (1995). J. Mol. Biol. 245, 324-330), provides a robust description of the insulin hexamer.

The amide rotational barrier in isonicotinamide: Dynamic NMR and ab initio studies.

We report use of dynamic nuclear magnetic resonance (NMR) to measure the amide rotational barrier in isonicotinamide. A significant challenge to obtaining good transition rates from dynamic NMR data is suppression of errors due to inherent line widths associated with transverse relaxation. We address this challenge with a fitting procedure that incorporates transverse relaxation over the temperature range of interest simply and reliably. The fitting model is nonlinear in only one of the fit parameters, namely, the activation enthalpy. This reduces parameter estimation to solution of a single transcendental equation, which avoids both a fine search over a multidimensional parameter space and extrapolation of a "limiting line width" solely from slow-exchange data. The activation enthalpy Delta H++ measured for isonicotinamide, +14.1 +/- 0.2 kcal/mol, falls between those of its regioisomers picolinamide and nicotinamide, which were reported in an earlier study. In that study, ab initio calculations of the rotational barriers helped to discern the relative importance of steric, electronic, and hydrogen-bonding effects in this biochemically significant combination of pyridine-ring and carboxamide moieties. A direct comparison between isonicotinamide and nicotinamide, where steric and hydrogen-bonding effects differ only slightly, permits a closer study of electronic considerations.

The amide rotational barriers in picolinamide and nicotinamide: NMR and ab initio studies.

Pyridine carboxamides are a class of medicinal agents with activity that includes the reduction of iron-induced renal damage, the regulation of nicotinamidase activity, and radio- and chemosensitization. Such pharmacological activities, and the prevalence of the carboxamide moiety and the importance of amide rotations in biology, motivate detailed investigation of energetics in these systems. In this study, we report the use of dynamic nuclear magnetic resonance to measure the amide rotational barriers in the pyridine carboxamides picolinamide and nicotinamide. The activation enthalpies and entropies of DeltaH++ = 12.9 +/- 0.3 kcal/mol and DeltaS++ = -7.7 +/- 0.9 cal/mol K for nicotinamide and DeltaH++ = 18.3 +/- 0.4 kcal/mol and DeltaS++ = +1.3 +/- 1.0 cal/mol K for picolinamide report a substantial energetic difference for these regioisomers. Ab initio calculations of the rotational barriers are in good agreement with the experimentally determined values and help partition the 5.4 kcal/mol enthalpy difference into its major contributions. Of principal importance are the variations in steric interactions in the ground states of picolinamide and nicotinamide, superior pi electron donation from the pyridine ring in the transition state of nicotinamide, and an intramolecular hydrogen bond in the ground state of picolinamide.