Ductal carcinoma of the breast: morphological aspects according to the age / Carcinoma ductal da mama: aspectos morfológicos de acordo com a idade

ABSTRACTIntroduction:Breast cancer is the most common cancer in women worldwide, and its morphological characteristics, despite the current molecular classification, also provide important information about the pattern of this disease.Objective:To analyze the morphological characteristics of invasive ductal breast carcinoma among women older and younger than 50 years.Method:302 cases of invasive ductal carcinoma patients treated at the Division of Mastology of the Universidade Federal do Ceará, in the period 2005-2014, aged ≤ 50 years and older. The following morphological characteristics were analyzed: larger tumor diameter (TD), lymph node metastasis (MX), and histological grade (HG).Results:The mean age of patients was 55.6 years. The average tumor size was 3.4 cm, 40% of tumors have diameter ≤ 2 cm and 60% > 2 cm. As for the histological grade, 23.7% were grade 1, 32.1% grade 2, and 42% grade 3. Axillary metastasis were present in 66% of the cases and 34% did not. Women aged ≤ 50 years had fewer tumors grade 1 (p = 0.002), compared with grades 2 and 3. Women older than 50 years had more grade 3 tumors (p = 0.002), and more tumors larger than 2 cm diameter (p < 0.001). The presence of metastasis predominated in both age groups when analyzed separately (p < 0.001).Conclusion:Women older than 50 years had larger and more morphologically undifferentiated tumors. Women aged ≤ 50 years had less well-differentiated tumors. There were no differences in morphology between these two age groups when compared each other.

RESUMOIntrodução:O câncer de mama é o câncer que mais acomete mulheres no mundo, e suas características morfológicas, a despeito da atual classificação molecular, ainda fornecem informações importantes sobre o comportamento desta doença.Objetivo:Analisar as características morfológicas dos carcinomas ductais invasivos da mama em mulheres acima e abaixo de 50 anos de idade.Métodos:Trezentos e dois casos de carcinoma ductal de pacientes do serviço de Mastologia da Universidade Federal do Ceará (UFC), no período de 2005 a 2014, com idade ≤ 50 anos e acima de 50 anos. Foram analisadas as seguintes características morfológicas: maior diâmetro do tumor (DT), presença de metástase axilar (MX) e grau histológico (GH).Resultados:A idade média das pacientes foi de 55,6 anos; o tamanho médio dos tumores, de 3,4 cm. Quarenta por cento dos tumores possuíam diâmetro ≤ 2 cm e 60%, > 2 cm. Quanto ao grau histológico, 23,7% eram grau 1, 32,1%, grau 2 e 42%, grau 3. Sessenta e seis por cento dos casos apresentaram metástase axilar e 34% não. Mulheres com idade ≤ 50 anos apresentaram menos tumores grau 1 (p = 0,002) em relação aos graus 2 e 3. Mulheres acima de 50 anos apresentaram mais tumores grau 3 (p = 0,002) e mais tumores com mais de 2 cm de diâmetro (p < 0,001). A presença de metástase predominou nas duas faixas etárias quando analisadas isoladamente (p < 0,001).Conclusão:Mulheres acima de 50 anos apresentaram tumores maiores e de morfologia mais indiferenciada; mulheres com idade ≤ 50 anos, menos tumores bem diferenciados. Não houve diferença da morfologia entre as faixas etárias quando comparadas entre si.

Carrier-controlled ferromagnetism in transparent oxide semiconductors.

The search for an ideal magnetic semiconductor with tunable ferromagnetic behaviour over a wide range of doping or by electrical gating is being actively pursued as a major step towards realizing spin electronics. A magnetic semiconductor having a high Curie temperature, capable of independently controlled carrier density and magnetic doping, is crucial for developing spin-based multifunctional devices. Cr-doped In(2)O(3) is such a unique system, where the electrical and magnetic behaviour-from ferromagnetic metal-like to ferromagnetic semiconducting to paramagnetic insulator-can be controllably tuned by the defect concentration. An explicit dependence of magnetic interaction leading to ferromagnetism on the carrier density is shown. A carrier-density-dependent high Curie temperature of 850-930 K has been measured, in addition to the observation of clear magnetic domain structures in these films. Being optically transparent with the above optimal properties, Cr-doped In(2)O(3) emerges as a viable candidate for the development of spin electronics.

Diagnosis of influenza virus: coming to grips with the molecular era.

Influenza viruses continually circulate and cause yearly epidemics, which kill 20,000 people in an average year in the United States. Occasionally and unpredictably, pandemic influenza strains sweep the world, infecting 20% to 40% of the world's population in a single year. In 1918, the worst influenza pandemic on record caused 675,000 deaths in the United States and up to 40 million deaths worldwide. Despite the prevalence of this virus, molecular assays for influenza diagnosis, surveillance, vaccine strain selection, and research have lagged behind such assays for other common viral pathogens. The extreme genetic variability of influenza viruses makes the design of useful molecular-based assays challenging, but several different approaches have been successfully used. RT-PCR is effective for the initial diagnosis and has greater sensitivity than other available rapid assays. Molecular assays also can be used to subtype influenza isolates, and sequence analysis of hemagglutinin may assist greatly in surveillance studies and vaccine strain selection. RT-PCR for influenza also can be performed from tissue biopsy specimens for both retrospective diagnosis and research.

A practical method for simultaneously determining the effective burst sizes and cycle times of viruses.

We describe combined analytic and experimental methods for determining reproductive statistics from time-series data. Our computational methods derive four fundamental measures from laboratory experiments: (i) average number of viral daughters; (ii) mean viral cycle time; (iii) standard deviation of the viral cycling time; and (iv) viral doubling time. Taken together, these four reproductive statistics characterize "age-specific fertility," a quantity that provides complete information on the reproduction of the average viral particle. In this paper, we emphasize applications relating to HIV and experiments for assessing cellular tropism, viral phenotypes, antiviral drugs, humoral immunity, and cytotoxic cellular immunity. Nevertheless, our method is quite flexible and applicable to the evaluation of drugs against bacterial, fungal, and parasitic infections, antineoplastic agents against cancer cells, and perturbations involving pest and wildlife releases in ecosystems.

L-arginine administration normalizes pressure natriuresis in hypertensive Dahl rats.

A blunted pressure-natriuretic response characterizes hypertension in the Dahl salt-sensitive rat. Long-term L-arginine administration prevents hypertension in these animals. To determine if long-term L-arginine corrects the pressure-natriuretic response, we gave salt-sensitive rats on an 8% sodium diet L-arginine or vehicle daily for 3 weeks. Identically treated salt-resistant rats served as controls. After 3 weeks, acute pressure-natriuresis curves were determined. To control for hypertension-induced renal damage, we also examined pressure natriuresis in salt-sensitive rats after short-term L-arginine. Baseline mean arterial pressure was 158 +/- 3 mm Hg in vehicle-treated salt-sensitive rats and 127 +/- 3 mm Hg in chronically L-arginine-treated salt-sensitive rats. During alterations in perfusion pressure, renal blood flow was autoregulated in all groups. Glomerular filtration rate was autoregulated in salt-resistant rats and L-arginine-treated salt-sensitive rats but fell with decreasing pressure in vehicle-treated salt-sensitive rats. Sodium excretion was greater (P < .05) in L-arginine-treated than in vehicle-treated salt-sensitive rats and did not differ from salt-resistant rats at 100, 125, and 158 mm Hg. The slope of the pressure-natriuresis relation was greater (P < .05) in chronically L-arginine-treated than in vehicle-treated salt-sensitive rats. L-Arginine had no effect on natriuresis in salt-resistant rats. Thus, long-term L-arginine administration normalizes pressure-natriuretic responses in salt-sensitive rats. The effect is not due to the prevention of renal damage and is specific to the salt-sensitive strain.(ABSTRACT TRUNCATED AT 250 WORDS)

Factors underlying spontaneous inactivation and susceptibility to neutralization of human immunodeficiency virus.

To determine the factors governing inactivation and neutralization, physical, chemical, and biological assays were performed on a molecular clone of human immunodeficiency type 1 (HIV-1HXB3). This included quantitative electron microscopy, gp120 and p24 enzyme-linked immunosorbent assays, reverse, transcriptase assays, and quantitative infectivity assays. For freshly harvested stocks, the ratio of infectious to noninfectious viral particles ranged from 10(-4) to 10(-7) in viral stocks containing 10(9) to 10(10) physical particles per milliliter. There were relatively few gp120 knobs per HIV particle, mean approximately 10 when averaged over the total particle count. Each HIV particle contained a mean approximately 5 x 10(-17) g of p24 and approximately 2 x 10(-16) g of RNA polymerase, corresponding to about 1200 and 80 molecules, respectively. The spontaneous shedding of gp120 envelope proteins from virions was exponential, with a half-life approximately 30 hr. The loss of RNA polymerase activity in virons was also exponential, with a half-life approximately 40 hr. The physical breakup of virions and the dissolution of p24 core proteins were slow (half-life greater than 100 hr) compared to the gp120 shedding and polymerase loss rates. The decay of HIV-1 infectivity was found to obey superimposed single- and multihit kinetics. At short preincubation times, the loss of infectivity correlated with spontaneous shedding of gp120 from virions. At longer times, an accelerating decay rate indicated that HIV requires a minimal number of gp120 molecules for efficient infection of CD4+ cells. The blocking activity of recombinant soluble CD4 (sCD4) and phosphonoformate (foscarnet) varied with the number of gp120 molecules and number of active RNA polymerase molecules per virion, respectively. These results demonstrate that the physical state of virions greatly influences infectivity and neutralization. The knowledge gained from these findings will improve the reliability of in vitro assays, enhance the study of wild-type strains, and facilitate the evaluation of potential HIV therapeutics and vaccines.

The auto-regulation model: a unified concept of how HIV regulates its infectivity, pathogenesis and persistence.

The life cycle of HIV can be divided into two distinct stages: intracellular and extracellular. The prevailing view is that the intracellular stage provides the only locus for regulating the virus in response to physiologic stimuli. Such regulation is accomplished by modulating the rates of transcription, translation and viral assembly. The extracellular stage consists of physical processes such as diffusion, adhesion and penetration of cells by viral particles. These latter processes are commonly thought to be "automatic" and not subject to regulation. For the past several years, we have developed means of more carefully measuring and characterizing the extracellular stage of HIV infection, and we have obtained evidence indicating that novel regulatory processes do, in fact, take place during this extracellular stage. We believe that this extracellular regulation permits HIV to adapt to a wide range of physiologic cell densities, to maintain persistent but slow growing infection, and to defeat the protective activity of humoral blockers. The overall purpose of this review is to consider our evidence for this hypothesis.

Blocking of human immunodeficiency virus infection depends on cell density and viral stock age.

Quantitative infectivity assays were used to study how the blocking activity of soluble CD4 (sCD4) is affected by sCD4 concentration, target cell density, and viral stock age. During incubation with 20 nM sCD4, human immunodeficiency virus type 1 (HIV-1) stocks underwent irreversible inactivation. In contrast, inactivation with 2 nM sCD4 was almost entirely reversible. At lower sCD4 concentrations (less than or equal to 2 nM) and target cell densities of 6.25 x 10(4) ml-1, sCD4 blocking activity for HIV-1 gave a gp120-sCD4 association constant (Kassoc) of 1.7 x 10(9) M-1, which agrees with chemical measurements. At the higher density of 1.6 x 10(7) cells ml-1, however, the blocking activity was 20-fold less. During incubation of HIV-1 stock optimized for infectivity by rapid harvest, sCD4 blocking activity increased 20-fold during a 3-h window. These results show that competitive blocking activity depends strongly on target cell density and virion age. Thus, unappreciated variations in HIV stocks and assay conditions may hinder comparisons of blockers from laboratory to laboratory, and the age of HIV challenge stocks may influence studies of drug and vaccine efficacy. The results also suggest that blocking of viral particles in lymphoid compartments will require very high competitive blocker concentrations, which may explain the refractory outcomes from sCD4-based drug trials in humans.

HIV requires multiple gp120 molecules for CD4-mediated infection.

Binding of glycoprotein gp120 to the T cell-surface receptor CD4 is a crucial step in CD4-dependent infection of a target cell by the human immunodeficiency virus (HIV). Blocking some or all gp120 molecules on the viral surface should therefore inhibit infection. Consequently, competitive receptor inhibitors, such as soluble synthetic CD4 (sCD4), synthetic CD4 peptides and immunoglobulins, have been investigated in vitro and in vivo, but little is known about the molecular mechanisms of these inhibitors. We have now quantitatively examined blocking by soluble CD4 in the hope of gaining insight into the complex process of viral binding, adsorption and penetration. At low sCD4 concentrations, the inhibition in three HIV strains is proportional to the binding of gp120. The biological association constant (gp120-sCD4 Kassoc) for HIV-2NIHZ is (8.5 +/- 0.5) x 10(7) M-1, whereas Kassoc for HIV-1HXB3 (1.4 +/- 0.2) and HIV-1MN (1.7 +/- 0.1) x 10(9) M-1 are 15-20-fold larger. For all three viral strains, the biological Kassoc from infectivity assays is comparable to the chemical Kassoc. The inhibitory action of sCD4 at high concentrations, however, is not fully explained by simple proportionality with the binding to gp120. Positive synergy in blocking of infection occurs after about half the viral gp120s molecules are occupied, and is identical for all three viral strains, despite the large differences in Kassoc. Our method of measuring the viral-cell receptor Kassoc directly from infectivity assays is applicable to immunoglobulins, to other viruses and to assays using primary or transformed cell lines.

Coronary microvascular responses to reductions in perfusion pressure. Evidence for persistent arteriolar vasomotor tone during coronary hypoperfusion.

The goals of this study were to test the following hypotheses: 1) Coronary autoregulatory adjustments to decreases in perfusion pressure occur primarily in coronary arterioles (less than 150 microns in diameter). 2) Small coronary arteries (greater than 150 microns in diameter) can be recruited to participate in the autoregulatory adjustments as perfusion pressure is progressively lowered. 3) Small arterioles are the location of vasodilator reserve in the coronary microcirculation during hypoperfusion. Studies were performed in anesthetized open-chest dogs in which coronary perfusion pressures were reduced to 80, 60, 40, and 30 mm Hg. During reductions in coronary perfusion pressure, measurements were made of systemic hemodynamics, myocardial blood flow (radioactive microspheres), and coronary microvascular diameters. Arterial pressure and heart rate were largely unchanged during the experimental maneuvers. Measurements of microvascular diameters in the beating heart were performed during epi-illumination via a stroboscopic light source synchronized to the cardiac cycle using fluorescence intravital microscopy. Coronary autoregulatory adjustments were evident during reductions in perfusion pressure from control (96 mm Hg) to 80 and 60 mm Hg. Blood flow was unchanged from control, and active vasodilation of coronary arterioles was observed. At 80 mm Hg, only coronary arterioles dilated (4.4 +/- 1.2%), whereas at 60 mm Hg both small arteries (4.9 +/- 2.2%) and arterioles (6.9 +/- 1.2%) demonstrated significant vasodilation (p less than 0.05). The magnitude of dilation (i.e., percent increase in diameter) was inversely related to the initial diameter; that is, the arterioles dilated to a greater extent, percentage wise, than the small arteries. At 40 mm Hg, myocardial blood flow decreased slightly from that under control conditions, but coronary arterioles dilated to a greater extent than at 60 mm Hg (8.1 +/- 1.6%); yet, microvessels were incompletely vasodilated, because adenosine produced a further increase in microvessel diameter (12.5 +/- 2.1%) (p less than 0.05). At a perfusion pressure of 30 mm Hg, arterioles demonstrated a decrease in vascular diameter (-0.2 +/- 2.1%), which was reversed by adenosine (11.1 +/- 3.1%). From these results we concluded the following: 1) Coronary autoregulatory adjustments involve primarily coronary arteriolar vessels, but small coronary arteries can be recruited to participate in the autoregulatory response. 2) The magnitude of vessel dilation appears to be inversely related to vascular diameter. 3) Coronary arterioles are not maximally vasodilated during coronary hypoperfusion, and these vessels may be the source of persistent vasomotor tone during coronary insufficiency.

Microvascular occlusions promote coronary collateral growth.

The objective of this study was to examine whether myocardial ischemia without alterations in pressure gradients between large epicardial coronary arteries was a sufficient stimulus to produce coronary collateral growth and development. To accomplish this aim, we partially embolized the circumflex coronary perfusion territory with 25-microns diameter microspheres to produce multiple microvascular occlusions, sufficient to abolish or greatly attenuate coronary vasodilator reserve. The embolization procedure was performed in two groups of dogs during aseptic surgery. After the dogs recovered for 1-3 wk (short-term embolization) or 6-8 wk (long-term embolization), indexes of vascular growth were compared with a group of control animals in which all operative procedures were performed, except embolization. Retrograde blood flow, an index of collateral blood flow and coronary vascular resistance, was determined in an isolated beating empty heart preparation during coronary vasodilation with adenosine. Circumflex retrograde blood flow from the left anterior descending artery was increased from 0.09 ml.min-1.g-1 (sham) to 0.21 and 0.17 ml.min-1.g-1 in the short-term and long-term groups, respectively (P less than 0.05). Collateral blood flow from the septal artery was also increased from 0.03 ml.min-1.g-1 (sham) to 0.08 ml.min-1.g-1 (P less than 0.05) in the short-term group. Collateral contribution from the right coronary artery was not significantly altered in either group of embolization animals. The contributions of epicardial and intramyocardial collaterals to the total retrograde flow were also determined and were found to be different among the three experimental groups.(ABSTRACT TRUNCATED AT 250 WORDS)

Effects of atherosclerosis on the coronary microcirculation.

We tested the hypothesis that atherosclerosis potentiates coronary vasoconstriction to serotonin and ergonovine. Coronary microvascular pressures and diameters were measured in the beating left ventricle in normal and atherosclerotic cynomolgus monkeys. Pressures were measured in arteries (190-350 microns diam) that were distal to atherosclerotic lesions. Microvascular pressure and simultaneous measurements of aortic pressure and myocardial blood flow were used to calculate segmental vascular resistance (large artery resistance and microvascular resistance) during serotonin, phenylephrine, and ergonovine dosages. Aortic pressure was maintained constant during all interventions. Administration of phenylephrine (50 micrograms.kg-1.min-1 iv) produced a similar increase in microvascular resistance from base line (P less than 0.05) in atherosclerotic and normal animals, 26 +/- 5 and 14 +/- 9 mmHg.min.g.ml-1, respectively. Serotonin (50 micrograms/min) did not influence coronary resistance in normal animals but produced a significant increase in both large artery (8 +/- 3 mmHg.min.g.ml-1) and microvascular resistance (21 +/- 6 mmHg.min.g.ml-1) in atherosclerotic animals (P less than 0.05). A higher dose of serotonin (200 micrograms/min) produced a modest increase in large artery resistance from base line in normal animals (3 +/- 1 mmHg.min.g.ml-1) and a greater increase in atherosclerotic animals (9 +/- 4 mmHg.min.g.ml-1) (P less than 0.05 vs. normals). Ergonovine (10 micrograms.kg-1.min-1 iv) elevated microvascular resistance in both normal and atherosclerotic animals (P less than 0.05) but increased large artery resistance only in atherosclerotic animals (10 +/- 4 mmHg.min.g.ml-1) (P less than 0.05). In summary, coronary vasoconstrictor responses to serotonin and ergonovine were potentiated by atherosclerosis. Because augmented constrictor responses to serotonin were observed in both the diseased arteries and the microcirculation of atherosclerotic animals, we speculate that the pathophysiological consequences of atherosclerosis extend into the microcirculation.

Risk behavior-based model of the cubic growth of acquired immunodeficiency syndrome in the United States.

The cumulative number of cases of acquired immunodeficiency syndrome (AIDS) in the United States has grown as the cube of time rather than exponentially. We explain this by interactions involving partner choice and sexual frequency in a risk-behavior model with biased mixing. This leads to a saturation wave of infection moving from high- to low-risk groups. If this description is correct, then the decreasing growth rate of AIDS cases is not due to behavior changes; rather it is due to the intrinsic epidemiology of the disease.

Quantifying the infectivity of human immunodeficiency virus.

We have developed a mathematical model that quantifies lymphocyte infection by human immunodeficiency virus (HIV) and lymphocyte protection by blocking agents such as soluble CD4. We use this model to suggest standardized parameters for quantifying viral infectivity and to suggest techniques for calculating these parameters from well-mixed infectivity assays. We discuss the implications of the model for our understanding of the infectious process and virulence of HIV in vivo.

Redistribution of coronary microvascular resistance produced by dipyridamole.

This study assessed the redistribution of coronary microvascular resistance during vasodilation produced by dipyridamole. Measurements of microvascular diameter and pressure in the beating left ventricle of anesthetized cats were accomplished by means of a computer-controlled system that enabled measurements in the beating heart. Resistances of coronary arteries, microvessels, and veins were calculated from the quotients of the pressure gradient across each vascular compartment and myocardial perfusion (radioactive microspheres). Administration of dipyridamole increased coronary blood flow from 1.80 +/- 0.09 to 6.42 +/- 0.31 ml.min-1. g-1 (P less than 0.05). During control conditions, 25 +/- 8% of total resistance occurred in coronary arteries (proximal to 170 microns), 68 +/- 8% of total resistance was in coronary microvessels (between arterioles less than 170 microns in diameter and venules less than 150 microns in diameter), and 7 +/- 7% of resistance resided in veins (distal to 150 microns). There was a significant redistribution (P less than 0.05) of resistance in all vessel classes after dipyridamole: coronary arteries constituted 42 +/- 6%, microvessels contained 27 +/- 5%, and veins had 31 +/- 8% of total coronary resistance. During control conditions, vascular resistance in coronary arteries and microvessels was 17 +/- 4 and 45 +/- 6 mmHg.min.g.ml-1, respectively. During vasodilation, resistance was significantly reduced (P less than 0.05) in both the arterial and microvessel segments to 6 +/- 2 and 4 +/- 2 mmHg.min.g.ml-1, respectively. Venous resistance was not significantly affected during dipyridamole-induced vasodilation. In conclusion, there was a marked reduction of coronary vascular resistance in response to dipyridamole, with the major component accounted for by dilation of microvessels.(ABSTRACT TRUNCATED AT 250 WORDS)

Heterogeneous microvascular coronary alpha-adrenergic vasoconstriction.

We tested the hypothesis that humoral or neurogenic alpha-adrenergic activation in the coronary circulation would produce heterogeneous vascular reactions. To accomplish this, the epicardial coronary microcirculation was viewed through an intravital microscope using stroboscopic epi-illumination. Microvascular diameters were measured under control conditions during beta-adrenergic blockade (propranolol 1 mg/kg) and beta-adrenergic blockade with pacing; during coronary alpha-adrenergic activation in the presence of beta-adrenergic blockade with three doses of norepinephrine infusion (0.1, 0.5, and 1.0-2.0 micrograms/kg/min) or three frequencies of bilateral stellate nerve stimulation (2, 10, and 20 Hz); and during combined alpha- and beta-adrenergic blockade (phentolamine 2 mg/kg and propranolol 1 mg/kg). Diameters of both arterial and venous vessels were reduced during beta-adrenergic blockade but returned back to baseline with pacing. At the lowest level of norepinephrine infusion or frequency of bilateral stellate stimulation, microvessel constriction was not observed. At the higher doses of norepinephrine a -5.1 +/- 0.9% (1.0-2.0 micrograms/kg/min) and a -4.0 +/- 1.1% (0.5 micrograms/kg/min) decrease in diameter of arterial vessels greater than 100 microns in diameter were observed (p less than 0.05). At 10 Hz and 20 Hz of stellate stimulation, diameter decreased by -4.8 +/- 1.9% and -4.4 +/- 2.1%, respectively, in these relatively large vessels. Small coronary arterioles (less than 100 microns diameter) dilated significantly during the highest levels of nerve stimulation (9.2 +/- 2.5% increase in diameter) or infusion rate of norepinephrine (13.6 +/- 2.7% increase in diameter) (p less than 0.05). These constrictor and dilator responses were abolished following combined alpha- and beta-adrenergic blockade. Norepinephrine infusion resulted in a decrease in diameter of coronary veins and venules (7.2 +/- 1.3%) (p less than 0.05), whereas stellate stimulation did not significantly reduce venous and venular diameters. In summary, the coronary venous and venular vasculature responds to alpha-adrenergic activation from circulating norepinephrine but is not affected by stellate stimulation. In contrast, stellate stimulation and norepinephrine infusion elicit similar responses in the coronary arterial and arteriolar microvasculature. Constriction occurs in vessels greater than 100 microns in diameter, whereas dilation predominates in vessels less than 100 microns in diameter. Such heterogeneous arterial responses would undoubtedly result in a redistribution of coronary vascular resistance toward larger coronary arteries and arterioles.

Analytic results for quantifying HIV infectivity.

In a separate paper, we developed a mathematical model describing HIV infection and used it to suggest experiments for quantifying characteristic viral parameters. In this paper we generalize the model to any well-mixed assay system. We also present complete and rigorous derivations of fundamental results needed for the design and analysis of HIV infectivity assays. The model is applicable to infectious agents with multiple receptors for their target cell (e.g. HIV, Epstein-Barr virus and Plasmodium), and to blockers (both reversible and irreversible), as long as blocker and target cells are the same diffusion compartment.