This paper derives a linear, first-order, partial differential field equation (a Dirac-like equation) in the geometric calculus of the geometric algebra $${\mathcal {G}}_{4,1}$$ that has free plane-wave solutions distinct from one another that correspond to the left and right chiral states of the electron and the neutrino. Besides the usual spacetime dependence of plane waves, the solutions have a multivector structure yielding a ladder of states with raising and lowering operators appropriate to electroweak theory and having an $$SU(2)_L$$ relationship among the chiral electron and neutrino states. The required Dirac-like equation in $${\mathcal {G}}_{4,1}$$ results from a systematic review of Dirac-like equations (i.e., first-order field equations whose solutions also satisfy the Klein–Gordon equation) in geometric algebras of lower dimension.

This paper has two objectives. The first is to explore the form and action of raising and lowering operators expressed in geometric algebra (GA). The second is to show how increasing the number of dimensions of Euclidean space from three to four opens a new avenue for understanding the chiral asymmetry of electroweak interactions. These explorations are guided by isomorphisms among groups represented in complex Clifford algebra, matrix algebra, and GA. With these isomorphisms, expressions for raising and lowering operators for electron and neutrino states in complex Clifford algebra are translated into GA and elaborated to include positrons and antineutrinos. This paper addresses such operators in the context of the electroweak sector of the SM utilizing (1) the GA $$\mathcal {G}_3$$ for the Hestenes–Dirac equation in a Euclidean lab frame, (2) $$\mathcal {G}_4$$ to introduce chiral asymmetry, and (3) $$\mathcal {G}_{4,1}$$ to express the electroweak fermion states of the first generation of the SM and demonstrate their SU(2) relationships.

Geometric algebra (GA) offers an intriguing approach to understanding the fields of the standard model (SM) of elementary particle physics. This paper examines a geometric view of electron and neutrino fields in the electroweak sector of the SM. These fields are related by the transformations of the \({SU(2)}\) Lie group, with generators customarily represented by the \({2\times 2}\) complex Pauli matrices. In \({\mathcal{G}_3}\), the GA of three-dimensional Euclidean space, the three unit basis vectors may be used to provide a more geometrically oriented representation of \({SU(2)}\). In fact, \({\mathcal{G}_3}\) is sometimes referred to as the Pauli algebra. However, a more general representation of the special unitary group \({SU(n)}\) in GA is in terms of generators that are compound (non-blade) bivectors in \({\mathcal{G}_{2n}}\), the GA of \({2n}\)-dimensional Euclidean space. Therefore, a natural approach to electroweak theory mathematically is to work with \({SU(2)}\) generators as compound bivectors in \({\mathcal{G}_4}\). This approach leads one to consider electroweak fields as multivector fields in \({\mathcal{G}_4}\) that are solutions of the Dirac equation in four spatial dimensions and one time dimension. This paper examines such multivector fields and offers a new point of view on chiral projection of \({\mathcal{G}_3}\) fields. It is shown that \({SU(2)}\) representation in \({\mathcal{G}_4}\) leads naturally to the singlet/doublet structure of the chiral electroweak fields.

This chapter describes the Deposition and Response in the Respiratory Tract (DARRT) model, an improved medical model of particle size effects for bioaerosol inhalation hazards. The DARRT model accounts for variations in human response caused by differences in particle size, in particular for coarse particles that may be present near an aerosol dissemination source and that may remain suspended long enough in an urban environment to expose large numbers of people. Plume dispersion calculations with the US Defense Threat Reduction Agency’s Hazard Prediction and Assessment Capability (HPAC) show that, contrary to common beliefs, particles with aerodynamic diameters as large as 20–40 µm can be a significant hazard tens of kilometers downwind of a release. DARRT predicts the probability of infection or injury from inhalation exposure to bioaerosols with diameters between 0.01 and 100 µm and describes the resulting medical impact. Many current models assume that only 1–5 µm “respirable” particles capable of reaching the pulmonary region of the respiratory system are of concern; however, we find that inclusion of coarse (>5 µm diameter) particle deposition is also important. Coarse particle deposition in the nose, mouth and throat may pose a substantial health risk.

: The goal of this effort was to develop a tool that is capable of assessing the effect of CBRN exposures on civilian populations and medical infrastructure in order to inform DoD planners at the NORTHCOM/Joint Task Force (JTF) level what additional resources might be required to support civilian authorities. DRACCM allows planners to import exposures, calculate time-dependent casualties, and to assess the beneficial effect of medical countermeasures including the negative effect of non-ideal or late treatment.

: This paper describes a toxicokinetic/dynamic model of percutaneous exposure to several vesicants. The model also predicts the efficacy of proposed medical countermeasures via a pharmacokinetic/dynamic model of scavenger, cell-cycle inhibitor, PARP inhibitor, protease inhibitor, and anti-inflammatory treatment regimens.

A treatment model for Pu-238/239 internal contamination by DTPA was developed to evaluate efficacy of various treatment regimens.The model estimates Pu deposition, absorption, distribution, retention, excretion, and response to DTPA after an inhalation exposure.It estimates the acute red bone marrow and lung doses and whole body effective dose as a function of time.Model results compare favorably with human and animal data.Outputs from the model include Pu deposition in the respiratory tract, distribution in tissue compartments over time with and without treatment, excretion rates, and radiation doses to critical organs.Calculations from the model may be used to analyze consequences of exposure to Pu and the effect of treatment.15.

: A treatment model for Am-241 internal contamination by DTPA was developed to evaluate efficacy of various treatment regimens. The model estimates Am deposition, absorption, distribution, retention, excretion, and response to DTPA after an inhalation exposure. It estimates the acute red bone marrow and lung doses and whole body effective dose as a function of time. Model results compare favorably with human and animal data. Outputs from the model include Am deposition in the respiratory tract, distribution in tissue compartments over time with and without treatment, excretion rates, and radiation doses to critical organs. Calculations from the model may be used to analyze consequences of exposure to Am-241 and the effect of treatment.

Anatomically accurate computational fluid dynamics (CFD) models of the nasal passages of an infant (6 months old, 1.3 kg) and adult (7 years old, 11.9 kg) rhesus monkey were used to predict nasal deposition of inhaled nano-and microparticles. Steady-state, inspiratory airflow simulations were conducted at flow rates equal to 100, 200 and 300% of the estimated minute volume for resting breathing in each model. Particle transport and deposition simulations were conducted using the Lagrangian method to track the motion of inhaled particles. Nasal deposition fractions were higher in the infant model than the adult model at equivalent physiologic flow rates. Deposition curves collapsed when differences in nasal geometry were accounted for by plotting microparticle deposition versus the Stokes number and nanoparticle deposition as a function of the Schmidt number and diffusion parameter. Particle deposition was also quantified on major nasal epithelial types. Maximum olfactory deposition ranged from 5 to 14% for 1-2 nm particles in the adult and infant models, depending on flow rate. For these particle sizes, deposition on respiratory/transitional epithelia ranged from 40 to 50%. Increased deposition was also predicted for olfactory and respiratory/transitional epithelia for particle sizes >5 mu m in the infant model and >8 mu m in the adult model. Semi-empirical curves were developed based on the CFD simulation results to allow for simplified calculations of age-based deposition in the rhesus monkey nasal passages that can be implemented into lung dosimetry models.

This paper explores the extension of the Legendre transform from scalar calculus to geometric calculus. In physics, the Legendre transform provides a change of variables to express equations of motion or other physical relationships in terms of the most convenient dynamical quantities for a given experimental or theoretical analysis. In classical mechanics and in field theory, the Legendre transform generates the Hamiltonian function of a system from the Lagrangian function or vice versa. In thermodynamics, the Legendre transform allows thermodynamic relationships to be written in terms of alternative sets of independent variables. In this paper, we review the properties of the Legendre transform in scalar calculus and show how an analogous transformation with similar properties may be constructed in geometric calculus.

: Hydrogen sulfide is similar in pathophysiology to cyanide in that it binds to mitochondrial cytochrome oxidase. By blocking oxidative respiration at the cellular level its effects are most evident in tissues with high metabolic demand such as the central nervous system, cardiovascular, and respiratory systems. The onset of signs and symptoms is rapid. Hydrogen sulfide also has local irritant effects, particularly upon the nose and membranes of the upper respiratory tract and the eyes. Physiological manifestations of H2S exposure may be quantified as a time-dependent severity vector described as increased time required to perform tasks owing to H2S-induced performance decrements as well as lethality. The model uses the time integral of agent concentration to calculate an internal dose, then determines the severity vector based on the dose. Outcome probability can be estimated as the time course till the casualty returns to duty, or else dies. Although H2S has not knowingly been used in warfare, it is a toxic industrial chemical/toxic industrial material that could be used as a terrorist agent.

Clifford analysis, particularly application of the geometric algebra of three-dimensional physical space and its associated geometric calculus, enables a compact formulation of Maxwell’s electromagnetic (EM) equations from a set of physically relevant and mathematically pleasing postulates. This formulation results in a natural extension of the Maxwell equations yielding wave solutions in addition to the usual EM waves. These additional solutions do not contradict experiment and have three properties in common with the apparent properties of dark energy. These three properties are that the wave solutions 1) propagate at the speed of light, 2) do not interact with ordinary electric charges or currents, and 3) possess retrograde momentum. By retrograde momentum, we mean that the momentum carried by such a wave is directed oppositely to the direction of energy transport. A “gas” of such waves generates negative pressure.

The Task-Taxon-Task method (Anno et al. DNA-TR-95-115, 1996) is a statistical modeling approach to predict performance decrements on behavioral tasks in response to various stressors. We describe the basics of the T3 method and our approach to adapting it to handle more acute stressors, which can require decomposition into task networks via logical or empirical analysis. We provide an illustrative example showing how the method can be used to account for performance decrements in manual tasks associated with wearing protective gloves. This illustration provides a substantive application in which the current T3 method can be augmented to account for performance decrements in a new sub-domain, while additionally providing lessons for extending the method to new stressors, performance domains, and behavior modeling systems.

Radiation exposure from Solar Particle Events (SPE) presents a significant health concern for astronauts for exploration missions outside the protection of the Earth's magnetic field, which could impair their performance and result in the possibility of failure of the mission. Assessing the potential for early radiation effects under such adverse conditions is of prime importance. Here we apply a biologically based mathematical model that describes the dose- and time-dependent early human responses that constitute the prodromal syndromes to consider acute risks from SPEs. We examine the possible early effects on crews from exposure to some historically large solar events on lunar and/or Mars missions. The doses and dose rates of specific organs were calculated using the Baryon radiation transport (BRYNTRN) code and a computerized anatomical man model, while the hazard of the early radiation effects and performance reduction were calculated using the Radiation-Induced Performance Decrement (RIPD) code. Based on model assumptions we show that exposure to these historical events would cause moderate early health effects to crew members inside a typical spacecraft or during extra-vehicular activities, if effective shielding and medical countermeasure tactics were not provided. We also calculate possible even worse cases (double intensity, multiple occurrences in a short period of time, etc.) to estimate the severity, onset and duration of various types of early illness. Uncertainties in the calculation due to limited data on relative biological effectiveness and dose-rate modifying factors for protons and secondary radiation, and the identification of sensitive sites in critical organs are discussed.