Mathematical modelling in nuclear medicine

Four decades of mapping and quantifying neuroreceptors at work in vivo by positron emission tomography

Decryption of brain images is the basis for the necessary translation of the findings from imaging to information required to meet the demands of clinical intervention. Tools of brain imaging, therefore, must satisfy the conditions dictated by the needs for interpretation in terms of diagnosis and prognosis. In addition, the applications must serve as fundamental research tools that enable the understanding of new therapeutic drugs, including compounds as diverse as antipsychotics, antidepressants, anxiolytics, and drugs serving the relief of symptoms from neurochemical disorders as unrelated as multiple sclerosis, stroke, and dementia. Here we review and explain the kinetics of methods that enable researchers to describe the brain’s work and functions. We focus on methods invented by neurokineticists and expanded upon by practitioners during decades of experimental work and on the methods that are particularly useful to predict possible future approaches to the treatment of neurochemical disorders. We provide an overall description of the basic elements of kinetics and the underlying quantification methods, as well as the mathematics of modeling the recorded brain dynamics embedded in the images we obtain in vivo. The complex presentation to follow is necessary to justify the contribution of modeling to the development of methods and to support the specifications dictated by the proposed use in clinical settings. The quantification and kinetic modeling processes are equally essential to image reconstruction and labeling of brain regions of structural or functional interest. The procedures presented here are essential tools of scientific approaches to all conventional and novel forms of brain imaging. The foundations of the kinetic and quantitative methods are keys to the satisfaction of clinicians that actively engage in treating the neurochemical disorders of mammalian brains in the fields of neurology, neurosurgery, and neuropsychiatry.

Automatic in-line quantitative myocardial perfusion mapping: Processing algorithm and implementation

PURPOSE

Quantitative myocardial perfusion mapping has advantages over qualitative assessment, including the ability to detect global flow reduction. However, it is not clinically available and remains a research tool. Building upon the previously described imaging sequence, this study presents algorithm and implementation of an automated solution for inline perfusion flow mapping with step by step performance characterization.

METHODS

Proposed workflow consists of motion correction (MOCO), arterial input function blood detection, intensity to gadolinium concentration conversion, and pixel-wise mapping. A distributed kinetics model, blood-tissue exchange model, is implemented, computing pixel-wise maps of myocardial blood flow (mL/min/g), permeability-surface-area product (mL/min/g), blood volume (mL/g), and interstitial volume (mL/g).

RESULTS

Thirty healthy subjects (11 men; 26.4 ± 10.4 years) were recruited and underwent adenosine stress perfusion cardiovascular MR. Mean MOCO quality score was 3.6 ± 0.4 for stress and 3.7 ± 0.4 for rest. Myocardial Dice similarity coefficients after MOCO were significantly improved (P < 1e-6), 0.87 ± 0.05 for stress and 0.86 ± 0.06 for rest. Arterial input function peak gadolinium concentration was 4.4 ± 1.3 mmol/L at stress and 5.2 ± 1.5 mmol/L at rest. Mean myocardial blood flow at stress and rest were 2.82 ± 0.47 mL/min/g and 0.68 ± 0.16 mL/min/g, respectively. The permeability-surface-area product was 1.32 ± 0.26 mL/min/g at stress and 1.09 ± 0.21 mL/min/g at rest (P < 1e-3). Blood volume was 12.0 ± 0.8 mL/100 g at stress and 9.7 ± 1.0 mL/100 g at rest (P < 1e-9), indicating good adenosine vasodilation response. Interstitial volume was 20.8 ± 2.5 mL/100 g at stress and 20.3 ± 2.9 mL/100 g at rest (P = 0.50).

CONCLUSIONS

An inline perfusion flow mapping workflow is proposed and demonstrated on normal volunteers. Initial evaluation demonstrates this fully automated solution for the respiratory MOCO, arterial input function left ventricle mask detection, and pixel-wise mapping, from free-breathing myocardial perfusion imaging.

Classic models for dynamic contrast-enhanced MRI

Dynamic contrast-enhanced MRI (DCE-MRI) is a functional MRI method where T1 -weighted MR images are acquired dynamically after bolus injection of a contrast agent. The data can be interpreted in terms of physiological tissue characteristics by applying the principles of tracer-kinetic modelling. In the brain, DCE-MRI enables measurement of cerebral blood flow (CBF), cerebral blood volume (CBV), blood-brain barrier (BBB) permeability-surface area product (PS) and the volume of the interstitium (ve ). These parameters can be combined to form others such as the volume-transfer constant K(trans) , the extraction fraction E and the contrast-agent mean transit times through the intra- and extravascular spaces. A first generation of tracer-kinetic models for DCE-MRI was developed in the early 1990s and has become a standard in many applications. Subsequent improvements in DCE-MRI data quality have driven the development of a second generation of more complex models. They are increasingly used, but it is not always clear how they relate to the models of the first generation or to the model-free deconvolution methods for tissues with intact BBB. This lack of understanding is leading to increasing confusion on when to use which model and how to interpret the parameters. The purpose of this review is to clarify the relation between models of the first and second generations and between model-based and model-free methods. All quantities are defined using a generic terminology to ensure the widest possible scope and to reveal the link between applications in the brain and in other organs.

Tracer kinetic modelling in MRI: estimating perfusion and capillary permeability

The tracer-kinetic models developed in the early 1990s for dynamic contrast-enhanced MRI (DCE-MRI) have since become a standard in numerous applications. At the same time, the development of MRI hardware has led to increases in image quality and temporal resolution that reveal the limitations of the early models. This in turn has stimulated an interest in the development and application of a second generation of modelling approaches. They are designed to overcome these limitations and produce additional and more accurate information on tissue status. In particular, models of the second generation enable separate estimates of perfusion and capillary permeability rather than a single parameter K(trans) that represents a combination of the two. A variety of such models has been proposed in the literature, and development in the field has been constrained by a lack of transparency regarding terminology, notations and physiological assumptions. In this review, we provide an overview of these models in a manner that is both physically intuitive and mathematically rigourous. All are derived from common first principles, using concepts and notations from general tracer-kinetic theory. Explicit links to their historical origins are included to allow for a transfer of experience obtained in other fields (PET, SPECT, CT). A classification is presented that reveals the links between all models, and with the models of the first generation. Detailed formulae for all solutions are provided to facilitate implementation. Our aim is to encourage the application of these tools to DCE-MRI by offering researchers a clearer understanding of their assumptions and requirements.

Pharmacokinetic analysis of tissue microcirculation using nested models: multimodel inference and parameter identifiability

The purpose of this study is to evaluate the identifiability of physiological tissue parameters by pharmacokinetic modeling of concentration-time curves derived under conditions that are realistic for dynamic-contrast-enhanced (DCE) imaging and to assess the information-theoretic approach of multimodel inference using nested models. Tissue curves with a realistic noise level were simulated by means of an axially distributed multipath reference model using typical values reported in literature on plasma flow, permeability-surface area product, and volume fractions of the intravascular and interstitial space. The simulated curves were subsequently analyzed by a two-compartment model containing these physiological quantities as fit parameters as well as by two reduced models with only three and two parameters formulated for the case of a permeability-limited and a flow-limited scenario, respectively. The competing models were ranked according to Akaike's information criterion (AIC), balancing the bias versus variance trade-off. To utilize the information available from all three models, model-averaged parameters were estimated using Akaike weights that quantify the relative strength of evidence in favor of each model. As compared to the full model, the reduced models yielded equivalent or even superior AIC values for scenarios where the structural information in the tissue curves on either the plasma flow or the capillary permeability was limited. Multimodel inference took effect to a considerable extent in half of the curves and improved the precision of the estimated tissue parameters. As theoretically expected, the plasma flow was subject to a systematic (but largely correctable) overestimation, whereas the other three physiological tissue parameters could be determined in a numerically robust and almost unbiased manner. The presented concept of pharmacokinetic analysis of noisy DCE data using three nested models under an information-theoretic paradigm offers promising prospects for the noninvasive quantification of physiological tissue parameters.

International Scientific Committee of Radionuclides in Nephrourology (ISCORN) consensus on renal transit time measurements

This report is the conclusion of the international consensus committee on renal transit time (subcommittee of the International Scientific Committee of Radionuclides in Nephrourology) and provides recommendations on measurement, normal values, and analysis of clinical utility. Transit time is the time that a tracer remains within the kidney or within a part of the kidney (eg, parenchymal transit time). It can be obtained from a dynamic renogram and a vascular input acquired in standardized conditions by a deconvolution process. Alternatively to transit time measurement, simpler indices were proposed, such as time of maximum, normalized residual activity or renal output efficiency. Transit time has been mainly used in urinary obstruction, renal artery stenosis, or renovascular hypertension and renal transplant. Despite a large amount of published data on obstruction, only the value of normal transit is established. The value of delayed transit remains controversial, probably due to lack of a gold standard for obstruction. Transit time measurements are useful to diagnose renovascular hypertension, as are some of the simpler indices. The committee recommends further collaborative trials.

In vivo assessment of cardiac insulin resistance by nuclear probes using an iodinated tracer of glucose transport

PURPOSE

Insulin resistance, implying depressed cellular sensitivity to insulin, is a risk factor for type 2 diabetes and cardiovascular disease. This study is the first step towards the development of a technique of insulin resistance measurement in humans with a new tracer of glucose transport, [(123)I]6-deoxy-6-iodo-D-glucose (6DIG).

METHODS

We investigated 6DIG kinetics in anaesthetised control rats and in three models of insulin-resistant rats: fructose fed, Zucker and ZDF. The study of myocardial 6DIG activity was performed under two conditions: first, 6DIG was injected under the baseline condition and then it was injected after a bolus injection of insulin. After each injection, radioactivity was measured over 45 min by external detection via NaI probes, in the heart and blood. A tri-compartment model was developed to obtain fractional transfer coefficients of 6DIG from the blood to the heart.

RESULTS

These coefficients were significantly increased with insulin in control rats and did not change significantly in insulin-resistant rats. The ratio of the coefficient obtained under insulin to that obtained under basal conditions gave an index of cardiac insulin resistance for each animal. The mean values of these ratios were significantly lower in insulin-resistant than in control rats: 1.16 +/- 0.06 vs 2.28 +/- 0.18 (p < 0.001) for the fructose-fed group, 0.92 +/- 0.05 vs 1.62 +/- 0.25 (p < 0.01) for the Zucker group and 1.34 +/- 0.06 vs 2.01 +/- 0.26 (p < 0.05) for the ZDF group.

CONCLUSION

These results show that 6DIG could be a useful tracer to image cardiac insulin resistance.

The A1 allele of the DRD2 gene (TaqI A polymorphisms) is associated with antisocial personality in a sample of alcohol-dependent patients

BACKGROUND

Presence of A1 allele of the DRD2 gene has been associated with a predisposition for alcoholism although there are limited data about its phenotypic expression in alcoholism.

OBJECTIVES

To determine the importance of the A1 allele in clinical variables of alcohol dependence.

METHODOLOGY

A sample of 103 alcohol-dependent males was studied. All patients were recruited consecutively from the general hospital and community settings. The diagnostics were made with the structured clinical interview for DSM-III-R (SCID); and the International Personality Disorder Examination (IPDE). Diagnosis of family alcoholism was made by direct interview or with the Research Diagnostic Criteria-Family History (RDC-FH). The Addiction Severity Index (ASI) and the Severity of Alcohol Dependence Scale (SADS) were used to assess alcohol dependence severity. Genotyping was done by polymerase chain reaction (PCR) and restriction fragment length polymorphism (RFLP) methods.

RESULTS

Approximately 39% of the sample carried the A1 allele (A1+ group). This group had higher prevalences of antisocial personality disorder (60% vs. 15.9%); and alcoholism family history (72.5% vs. 52.4%). Also A1+ had early onset alcohol abuse and more drinking problems. The presence of A1+ was the main factor to explain the diagnosis of antisocial personality disorder, but the weight of this factor was not sufficient to explain the complications assessed by the ASI.

CONCLUSIONS

Our results support the existence of an association between the A1 allele and factors resulting from dopaminergic deficiency, otherwise denominated reward deficiency syndrome.

Heterogeneity of cerebral blood flow in Alzheimer disease and vascular dementia

BACKGROUND AND PURPOSE

Alzheimer disease (AD) and vascular dementia (VaD) are the two major diseases that cause dementia, and early diagnosis is important. Single photon emission CT (SPECT) of cerebral blood flow (CBF) is used for the early detection of dementia and as an auxiliary method for follow-up. AD shows reduced posterior blood flow and VaD manifests reduced anterior blood flow on CBF SPECT images. We examined the usefulness of 3D fractal analysis of CBF SPECT images to objectively quantify the heterogeneity of CBF in patients with AD and VaD.

METHODS

Thirty-two patients with AD and 22 with VaD based on neuropsychologic tests and imaging findings, as well as 20 age-matched control subjects underwent technetium-99m hexamethyl propyleneamine oxime CBF SPECT. We then conducted statistical image processing by 3D fractal analysis on reconstructed data. Fractal dimension, an index of heterogeneity, was then calculated for the whole brain, as well as for the anterior and posterior regions of the brain. A higher fractal dimension indicates that the CBF SPECT image is uneven. The ratio of fractal dimension of the anterior region to fractal dimension of the posterior region (A/P ratio) was calculated. Heterogeneity of CBF was compared among the AD, VaD, and control groups.

RESULTS

Fractal dimensions of the AD, VaD, and control groups were 1.072+/-0.179 (mean +/- SD), 1.005+/-0.156, and 0.806+/-0.06, respectively. A significant difference of fractal dimension was noted between the control group and the two types of dementia (P<.0001); however, no significant difference was noted between the AD and VaD groups. The A/P ratios of the AD and VaD groups were significantly different (0.952 and 1.163, respectively; P<.01).

CONCLUSION

Analysis of CBF SPECT images quantitatively showed that the fractal dimension was significantly higher (indicating heterogeneity) in patients with AD and VaD when compared with age-matched control subjects. Comparison of the A/P ratio on CBF SPECT images between AD and VaD groups showed that the heterogeneity of CBF was posterior-dominant for AD and anterior-dominant for VaD. Thus, 3D fractal analysis enabled a simple and objective evaluation of the heterogeneity of CBF in patients with AD and VaD.

[A new method for quantification of hepatobiliary scintigraphy using 99mTc-mebrofenin. A comparative study]

A method based upon the application of mathematical techniques of deconvolution on the classical compartmental model for the quantitative study of liver function from hepatobiliary scintigraphy using 99mTc-mebrofenin data is proposed. The theory in which the method is based upon is presented and a comparison with a published methodology of obtaining the hepatic extraction after scintigraphic sudies has been performed using the results on 36 rats studies obtained with the two methods. A highly significant correlation between the two techniques was verified. The characteristics of the two methodologies, the proposed one based upon a theoretical approach and the other one on an empirical approximation are discussed. Comments are made on the interest and limitations of the presented technique that may be an useful tool for the evaluation of hepatic insufficiency.

Measurement of heterogeneous distribution on Technegas SPECT images by three-dimensional fractal analysis

This review article describes a method for quantifying heterogeneous distribution on Technegas (99mTc-carbon particle radioaerosol) SPECT images by three-dimensional fractal analysis (3D-FA). Technegas SPECT was performed to quantify the severity of pulmonary emphysema. We delineated the SPECT images by using five cut-offs (15, 20, 25, 30 and 35% of the maximal voxel radioactivity), and measured the total number of voxels in the areas surrounded by the contours obtained with each cut-off level. We calculated fractal dimensions from the relationship between the total number of voxels and the cut-off levels transformed into natural logarithms. The fractal dimension derived from 3D-FA is the relative and objective measurement, which can assess the heterogeneous distribution on Technegas SPECT images. The fractal dimension strongly correlate pulmonary function in patients with emphysema and well documented the overall and regional severity of emphysema.

[Multi-compartmental modelling and experimental design for glucose transport studies in insulin-resistant rats]

Many pathologies are associated with abnormalities of glucose metabolism or with perturbations of its transport (type 2 diabetes or insulin-resistance). The pre-diabetic state is characterised by a state of insulin-resistance, in others words a defect of glucose transport in insulin-sensible tissues, such as muscles and adipose tissues. The mathematical modelling of experimental data can be an excellent method to explore the mechanisms implied in the studied biological phenomenon. Thus, starting from a symbolic formulation like the compartmental modelling, it can be possible to develop a theoretical basis for the observation and to consider the best-adapted experiments for the study. We showed with mathematical models that [123I]-6-deoxy-6-iodo-D-glucose (6-DIG), shown as a tracer of glucose transport in vitro, could point out this transport abnormality. To quantify the insulin resistance, we estimated the fractional transfer coefficients of 6-DIG from the blood to the organs. We realised many studies to lead to a satisfying model; special attention has been paid to the precision of the parameter to select the best model. The results showed that by associating experimental data obtained with 6-DIG activities and an adapted mathematical model, discriminating parameters (in and out fractional transfer coefficients) between the two groups (control and insulin-resistant rats) could be pointed out.

Effect of flow and vascular heterogeneity on glucose metabolism in isolated dog hearts

In non-ischaemic myocardium glucose uptake is assumed to be proportional to blood flow. We investigated the effect of regional vascular heterogeneity on glucose metabolism at various flow rates in the isolated blood-perfused dog hearts. Aortic bolus injections contained an intravascular reference tracer (albumin) and two of three glucoses: L-glucose (an extracellular tracer), D-glucose and 2-deoxy-D-glucose. Flow ranged from 0.5 to 2 4 ml min(-1) g(-1). Vascular heterogeneity was calculated from the albumin outflow dilution curve. A three-region convection-diffusion from the abumin outflow dilution curve. A three-region, convection-diffusion model was fitted to outflow dilution curves to estimate glucose metabolic rate (consumption). The results of 2-deoxy-D-glucose experiments showed that the lumped constant was dependent on flow, glucose metabolic rate was proportional to flow and dependent on the heterogeneity of the myocardial vasculature. The results support the views that without accounting the regional flow heterogeneity, glucose metabolic rate will be underestimated.

The effect of ischemic injury on the cardiac transport of Tc-99m N-NOET in the isolated rabbit heart

BACKGROUND

Bis (N-ethoxy, N-ethyl dithiocarbamato) nitrido technetium-99m (V) (TcN-NOET) is a neutral lipophilic myocardial perfusion agent. The effect of ischemic injury on the cardiac transport of TcN-NOET and thallium-201 was determined in isolated rabbit hearts.

METHODS AND RESULTS

The multiple indicator dilution method was used to determine the maximum (Emax) and net extraction (Enet, at 5 minutes) of TcN-NOET and TI-201 at control and after 10 minutes (n = 4) or 45 minutes (n = 4) of no-flow ischemia. After 10 minutes of ischemia the mean Emax for T1-201 was unchanged, 0.86 +/- 0.03 vs 0.85 +/- 0.02, whereas TI-201 Enet showed a small decrease from 0.46 +/- 0.03 to 0.40 +/- 0.03, P < .001. Forty-five minutes of ischemia mildly reduced Emax for TI-201 (0.87 +/- 0.04 to 0.74 +/- 0.04, P < .001) and severely reduced Enet (0.46 +/-0.03 vs 0.16 +/- 0.04, P < .001). Neither Emax nor Enet for TcN-NOET was significantly affected by 10 minutes of ischemia (0.54 +/- 0.04 vs 0.58 +/- 0.03 and 0.24 +/- 0.04 vs 0.26 +/- 0.04, respectively). However, severe ischemic injury caused significant reductions versus control in both Emax (0.59 +/- 0.06 vs 0.42 +/- 0.05, P < .001) and Enet (0.27 +/- 0.03 vs 0.18 +/- 0.05, P < .01).

CONCLUSIONS

TcN-NOET is a new myocardial perfusion agent with moderate myocardial extraction. Although less sensitive than TI-201 to mild ischemic injury, TcN-NOET extraction and retention are decreased by severe ischemic injury, making uptake of TcN-NOET a possible marker of myocardial viability.

Kinetics of 111In-labeled bleomycin in patients with brain tumors: compartmental vs. non-compartmental models

The kinetics of an indium-111 labeled bleomycin complex (111In-BLMC) after rapid intravenous injection in patients with brain tumors was quantified by using compartmental and non-compartmental models. The models were applied to data obtained from 10 glioma, one meningioma, and one adenocarcinoma brain metastasis patients. Blood and urine samples from all the patients and tumor samples from three patients were collected. The mean transit time of 111In-BLMC in the plasma pool was 14 +/- 7 min without and 1.8 +/- 0.6 h when accounting for recirculation, and 13 +/- 4 h in the total body pool. The mean plasma clearance of 111In-BLMC was 0.3 +/- 0.1 m/blood/min and the mean half-life in urine was 3.5 +/- 0.6 h. The mean transfer coefficients for the open three-compartmental model were: excretion from plasma = 0.02 +/- 0.01, from depot to plasma = (12 +/- 9)*10(-4), from plasma to depot = 0.01 +/- 0.01, from tumor to plasma = 0.39 +/- 0.19 and from plasma to tumor = 1.11 +/- 0.57, all in units minute-1. The mean turnover time from the tumor was 4.5 +/- 2.7 min and from the depot 20 +/- 8 h. It is concluded that both compartmental and non-compartmental models are sufficient to describe the kinetics of indium-111 labeled bleomycin complex. The non-compartmental model is more practical and to some extent more efficient in describing the in vivo behaviors of 111In-BLMC than the compartmental model. The compartmental model used provides estimates of both extraction and excretion from the plasma and tumor.

Effects of acute ischemia and reperfusion on the myocardial kinetics of technetium 99m-labeled tetrofosmin and thallium-201

BACKGROUND

Effects of no-flow ischemia and reperfusion on myocardial extraction and retention of 99mTc-labeled tetrofosmin and 201Tl were investigated in seven isolated, blood-perfused rat hearts with isotope dilution studies at constant coronary perfusion.

METHODS AND RESULTS

After a control injection of tracers, no-flow ischemia was induced for 20 minutes. After coronary reflow, tracers were injected. Both maximal fractional extraction and capillary permeability-surface area product for tetrofosmin were significantly less than those for 201Tl (maximal fractional extraction 0.30 +/- 0.01 and 0.70 +/- 0.09, respectively, p < 0.001; capillary permeability-surface area product 0.66 +/- 0.14 and 2.29 +/- 0.61, respectively, p < 0.001). After no-flow ischemia-reperfusion, both maximal fractional extraction and capillary permeability-surface area product decreased for both tetrofosmin and 201Tl (decreases in maximal fractional extraction of 23% and 7%, respectively; decreases in capillary permeability-surface area product of 27% and 16%, respectively), although the difference reached statistical significance only for tetrofosmin. Net extraction at 5 minutes of both tracers decreased significantly after no-flow ischemia-reperfusion (tetrofosmin 20% decrease, p < 0.01; 201Tl 23% decrease, p < 0.02). Early (0 to 5 minutes) washout of tetrofosmin did not change after no-flow ischemia-reperfusion, whereas the 201Tl value increased significantly. Although late (5 to 19 minutes) washout of both tracers increased significantly after no-flow ischemia-reperfusion, the myocardial clearance rates for tetrofosmin were always significantly less than those noted for 201Tl.

CONCLUSIONS

The myocardial uptake of tetrofosmin is depressed (independent of blood flow) after severe ischemic injury, apparently resulting mainly from decreased transcapillary exchange. In contrast, the depressed uptake of 201Tl is related more to an accelerated early washout from injured myocardium than to a fairly stable initial transcapillary exchange.

Fractal analysis of striatal dopamine re-uptake sites

Spatial variation in regional blood flow, metabolism and receptor density within the brain and in other organs is measurable even with a low spatial resolution technique such as emission tomography. It has been previously shown that the observed variance increases with increasing number of subregions in the organ/tissue studied. This resolution-dependent variance can be described by fractal analysis. We studied striatal dopamine re-uptake sites in 39 healthy volunteers with high-resolution single-photon emission tomography using iodine-123 labelled 2beta-carbomethoxy-3beta-(4-iodophenyl)tropane ([123I]beta-CIT). The mean fractal dimension was 1.15+/-0.07. The results indicate that regional striatal dopamine re-uptake sites involve considerable spatial heterogeneity which is higher than the uniform density (dimension=1.00) but much lower than complete randomness (dimension=1.50). There was a gender difference, with females having a higher heterogeneity in both the left and the right striatum. In addition, we found striatal asymmetry (left-to-right heterogeneity ratio of 1.19+/-0.15; P<0.001), suggesting functional hemispheric lateralization consistent with the control of motor behaviour and integrative functions.

Nuclear medicine and mathematics

The purpose of this review is not to present a comprehensive description of all the mathematical tools used in nuclear medicine, but to emphasize the importance of the mathematical method in nuclear medicine and to elucidate some of the mathematical concepts currently used. We can distinguish three different areas in which mathematical support has been offered to nuclear medicine: physiology, methodology and data processing. Nevertheless, the boundaries between these areas can be indistinct. It is impossible in a single article to give even an idea of the extent and complexity of the procedures currently used in nuclear medicine, such as image processing, reconstruction from projections and artificial intelligence. These disciplines do not belong to nuclear medicine: they are already branches of engineering, and my interest will reside simply in revealing a little of the elegance and the fantastic potential of these new "allies" of nuclear medicine. In this review the mathematics of physiological interpretation and methodology are considered together in the same section. General aspects of data-processing methods, including image processing and artificial intelligence, are briefly analysed. The mathematical tools that are most often used to assist the interpretation of biological phenomena in nuclear medicine are considered; these include convolution and deconvolution methods, Fourier analysis, factorial analysis and neural networking.

Granulocyte margination in bone marrow: comparison with margination in the spleen and liver

The kinetics of radiolabelled granulocytes in the reticuloendothelial system were studied in order to evaluate granulocyte margination in bone marrow. A total of 34 patients took part in a two-part study. In the first part, bone marrow uptake of indium-111-labelled granulocytes was retrospectively analysed in early (3-h) and late (24-h) images in 26 patients, 13 with bronchiectasis and 13 with enclosed abdominal abscesses. The ratios between early and late counts from the bone marrow, spleen, liver and inflammatory lesion were used to quantify granulocyte margination in bone marrow, postulating that if the lesion to bone marrow ratio at 24 h exceeds the value at 3 h, then the "excess" bone marrow counts on the early images would represent margination. In the second part, this suggestion was prospectively tested using Rutland-Patlak graphical and deconvolution analysis of dynamic data, acquired in 8 patients undergoing routine scanning with technetium-99m HMPAO-labelled granulocytes. In the first part of the study, it appeared that the bone marrow is a regional site of granulocyte margination, like the spleen, with at least one-half of the 3-h marrow signal arising from marginated granulocytes, compared with about two-thirds from the spleen. In the second part, it was found that the gradient of the Patlak plot, based on spleen and marrow, continuously decreased, consistent with bi-directional movement of cells between these organs and the blood. Granulocyte pooling in the marrow was confirmed with deconvolution analysis, which generated biphasic retention functions for marrow and spleen. These curves were also consistent with two-way granulocyte exchange, and gave mean cell transit times in both organs of about 12 min and probabilities of extraction on each pass of 5-10%. We conclude that granulocytes marginate in bone marrow to an extent similar to that in the spleen.

Interaction of technetium 99m-labeled teboroxime with red blood cells reduces the compound's extraction and increases apparent cardiac washout

BACKGROUND

99mTc-labeled teboroxime shows high myocardial extraction in both in vivo animal and in vitro cell culture and isolated heart studies. Whereas in vivo studies show rapid myocardial clearance of teboroxime, in vitro cell culture and isolated heart studies show slower washout comparable to that of 201Tl. Binding of teboroxime to blood components may contribute to these conflicting results.

METHODS AND RESULTS

We measured teboroxime extraction in the isolated blood-perfused rabbit heart after injection in saline solution, brief incubation in red blood cell perfusate, or 4-hour incubation with human red blood cells. Teboroxime in saline solution showed high extraction (Emax = 0.89 +/- 0.02; Enet = 0.69 +/- 0.02), whereas brief incubation in perfusate (Emax = 0.60 +/- 0.06; Enet = 0.48 +/- 0.05) or prolonged incubation with human red blood cells (Emax = 0.43 +/- 0.09; Enet = 0.38 +/- 0.07) resulted in reduced extraction. Teboroxime clearance was similar for all groups and was slower than 201Tl clearance. Analysis of total residual cardiac teboroxime (comparable to external imaging) showed that teboroxime clearance was biexponential. Reduced extraction of teboroxime in red blood cells resulted in an increased size of the rapidly clearing (unextracted) fraction, giving the appearance of rapid myocardial washout.

CONCLUSIONS

Teboroxime has a high myocardial extraction. Binding to blood components reduces teboroxime extraction and increases the rate of cardiac teboroxime clearance.

Blood platelet kinetics in normal subjects modelled by compartmental analysis

The purpose of this study was to describe the function of platelets throughout their life span by expressing their in vivo distribution and kinetic behaviour in mathematical terms by using multicompartmental analysis. The distribution of indium-111 labelled platelets in five normal subjects was imaged and quantified with a scintillation camera image processing system. Serial blood samples were also obtained. The data were modelled using the SAAM (Simulation Analysis and Modelling) compartmental computer program. Five models were entertained to evaluate the role of platelets that were either functional or injured during collection and their interaction with the liver, spleen and vascular endothelium. Models were evaluated by comparing F values calculated from the least squares estimate obtained from each model. The Dornhorst function was used to describe the sequestration of platelets in the compartmental model. Results indicated that the data could not be satisfactorily simulated when compartments were included that simulated only functional and sequestered platelets (model 1). It was necessary to include compartments that simulated the kinetics of collection-injured platelets in the liver (model 2) and spleen (model 3). The model that simulated the interaction with the vascular endothelium (model 5) showed a visual but not significant improvement in the fitting of the observed data compared to model 3. The mean organ uptake and range indicated in parentheses were calculated at equilibrium. There were 20% (15%-27%) of the injected platelets in the spleen, 10% (8%-11%) in the liver and 70% (64%-75%) in the circulation.(ABSTRACT TRUNCATED AT 250 WORDS)