Mathematical modeling of brain metastases growth and response to therapies: A review.

Brain metastases (BMs) are the most common intracranial tumor type and a significant health concern, affecting approximately 10% to 30% of all oncological patients. Although significant progress is being made, many aspects of the metastatic process to the brain and the growth of the resulting lesions are still not well understood. There is a need for an improved understanding of the growth dynamics and the response to treatment of these tumors. Mathematical models have been proven valuable for drawing inferences and making predictions in different fields of cancer research, but few mathematical works have considered BMs. This comprehensive review aims to establish a unified platform and contribute to fostering emerging efforts dedicated to enhancing our mathematical understanding of this intricate and challenging disease. We focus on the progress made in the initial stages of mathematical modeling research regarding BMs and the significant insights gained from such studies. We also explore the vital role of mathematical modeling in predicting treatment outcomes and enhancing the quality of clinical decision-making for patients facing BMs.

Volumetric analysis: Rethinking brain metastases response assessment.

Background: The Response Assessment in Neuro-Oncology for Brain Metastases (RANO-BM) criteria are the gold standard for assessing brain metastases (BMs) treatment response. However, they are limited by their reliance on 1D, despite the routine use of high-resolution T1-weighted MRI scans for BMs, which allows for 3D measurements. Our study aimed to investigate whether volumetric measurements could improve the response assessment in patients with BMs. Methods: We retrospectively evaluated a dataset comprising 783 BMs and analyzed the response of 185 of them from 132 patients who underwent stereotactic radiotherapy between 2007 and 2021 at 5 hospitals. We used T1-weighted MRIs to compute the volume of the lesions. For the volumetric criteria, progressive disease was defined as at least a 30% increase in volume, and partial response was characterized by a 20% volume reduction. Results: Our study showed that the proposed volumetric criteria outperformed the RANO-BM criteria in several aspects: (1) Evaluating every lesion, while RANO-BM failed to evaluate 9.2% of them. (2) Classifying response effectively in 140 lesions, compared to only 72 lesions classified by RANO-BM. (3) Identifying BM recurrences a median of 3.3 months earlier than RANO-BM criteria. Conclusions: Our study demonstrates the superiority of volumetric criteria in improving the response assessment of BMs compared to the RANO-BM criteria. Our proposed criteria allow for evaluation of every lesion, regardless of its size or shape, better classification, and enable earlier identification of progressive disease. Volumetric criteria provide a standardized, reliable, and objective tool for assessing treatment response.

Chromosomal instability in aneuploid acute lymphoblastic leukemia associates with disease progression.

Chromosomal instability (CIN) lies at the core of cancer development leading to aneuploidy, chromosomal copy-number heterogeneity (chr-CNH) and ultimately, unfavorable clinical outcomes. Despite its ubiquity in cancer, the presence of CIN in childhood B-cell acute lymphoblastic leukemia (cB-ALL), the most frequent pediatric cancer showing high frequencies of aneuploidy, remains unknown. Here, we elucidate the presence of CIN in aneuploid cB-ALL subtypes using single-cell whole-genome sequencing of primary cB-ALL samples and by generating and functionally characterizing patient-derived xenograft models (cB-ALL-PDX). We report higher rates of CIN across aneuploid than in euploid cB-ALL that strongly correlate with intraclonal chr-CNH and overall survival in mice. This association was further supported by in silico mathematical modeling. Moreover, mass-spectrometry analyses of cB-ALL-PDX revealed a "CIN signature" enriched in mitotic-spindle regulatory pathways, which was confirmed by RNA-sequencing of a large cohort of cB-ALL samples. The link between the presence of CIN in aneuploid cB-ALL and disease progression opens new possibilities for patient stratification and offers a promising new avenue as a therapeutic target in cB-ALL treatment.

Radiation necrosis after radiation therapy treatment of brain metastases: A computational approach.

Metastasis is the process through which cancer cells break away from a primary tumor, travel through the blood or lymph system, and form new tumors in distant tissues. One of the preferred sites for metastatic dissemination is the brain, affecting more than 20% of all cancer patients. This figure is increasing steadily due to improvements in treatments of primary tumors. Stereotactic radiosurgery (SRS) is one of the main treatment options for patients with a small or moderate number of brain metastases (BMs). A frequent adverse event of SRS is radiation necrosis (RN), an inflammatory condition caused by late normal tissue cell death. A major diagnostic problem is that RNs are difficult to distinguish from BM recurrences, due to their similarities on standard magnetic resonance images (MRIs). However, this distinction is key to choosing the best therapeutic approach since RNs resolve often without further interventions, while relapsing BMs may require open brain surgery. Recent research has shown that RNs have a faster growth dynamics than recurrent BMs, providing a way to differentiate the two entities, but no mechanistic explanation has been provided for those observations. In this study, computational frameworks were developed based on mathematical models of increasing complexity, providing mechanistic explanations for the differential growth dynamics of BMs relapse versus RN events and explaining the observed clinical phenomenology. Simulated tumor relapses were found to have growth exponents substantially smaller than the group in which there was inflammation due to damage induced by SRS to normal brain tissue adjacent to the BMs, thus leading to RN. ROC curves with the synthetic data had an optimal threshold that maximized the sensitivity and specificity values for a growth exponent ß* = 1.05, very close to that observed in patient datasets.

Overcoming chemotherapy resistance in low-grade gliomas: A computational approach.

Low-grade gliomas are primary brain tumors that arise from glial cells and are usually treated with temozolomide (TMZ) as a chemotherapeutic option. They are often incurable, but patients have a prolonged survival. One of the shortcomings of the treatment is that patients eventually develop drug resistance. Recent findings show that persisters, cells that enter a dormancy state to resist treatment, play an important role in the development of resistance to TMZ. In this study we constructed a mathematical model of low-grade glioma response to TMZ incorporating a persister population. The model was able to describe the volumetric longitudinal dynamics, observed in routine FLAIR 3D sequences, of low-grade glioma patients acquiring TMZ resistance. We used the model to explore different TMZ administration protocols, first on virtual clones of real patients and afterwards on virtual patients preserving the relationships between parameters of real patients. In silico clinical trials showed that resistance development was deferred by protocols in which individual doses are administered after rest periods, rather than the 28-days cycle standard protocol. This led to median survival gains in virtual patients of more than 15 months when using resting periods between two and three weeks and agreed with recent experimental observations in animal models. Additionally, we tested adaptive variations of these new protocols, what showed a potential reduction in toxicity, but no survival gain. Our computational results highlight the need of further clinical trials that could obtain better results from treatment with TMZ in low grade gliomas.

The shape of cancer relapse: Topological data analysis predicts recurrence in paediatric acute lymphoblastic leukaemia.

Although children and adolescents with acute lymphoblastic leukaemia (ALL) have high survival rates, approximately 15-20% of patients relapse. Risk of relapse is routinely estimated at diagnosis by biological factors, including flow cytometry data. This high-dimensional data is typically manually assessed by projecting it onto a subset of biomarkers. Cell density and "empty spaces" in 2D projections of the data, i.e. regions devoid of cells, are then used for qualitative assessment. Here, we use topological data analysis (TDA), which quantifies shapes, including empty spaces, in data, to analyse pre-treatment ALL datasets with known patient outcomes. We combine these fully unsupervised analyses with Machine Learning (ML) to identify significant shape characteristics and demonstrate that they accurately predict risk of relapse, particularly for patients previously classified as 'low risk'. We independently confirm the predictive power of CD10, CD20, CD38, and CD45 as biomarkers for ALL diagnosis. Based on our analyses, we propose three increasingly detailed prognostic pipelines for analysing flow cytometry data from ALL patients depending on technical and technological availability: 1. Visual inspection of specific biological features in biparametric projections of the data; 2. Computation of quantitative topological descriptors of such projections; 3. A combined analysis, using TDA and ML, in the four-parameter space defined by CD10, CD20, CD38 and CD45. Our analyses readily extend to other haematological malignancies.

Growth exponents reflect evolutionary processes and treatment response in brain metastases.

Tumor growth is the result of the interplay of complex biological processes in huge numbers of individual cells living in changing environments. Effective simple mathematical laws have been shown to describe tumor growth in vitro, or simple animal models with bounded-growth dynamics accurately. However, results for the growth of human cancers in patients are scarce. Our study mined a large dataset of 1133 brain metastases (BMs) with longitudinal imaging follow-up to find growth laws for untreated BMs and recurrent treated BMs. Untreated BMs showed high growth exponents, most likely related to the underlying evolutionary dynamics, with experimental tumors in mice resembling accurately the disease. Recurrent BMs growth exponents were smaller, most probably due to a reduction in tumor heterogeneity after treatment, which may limit the tumor evolutionary capabilities. In silico simulations using a stochastic discrete mesoscopic model with basic evolutionary dynamics led to results in line with the observed data.

A comprehensive dataset of annotated brain metastasis MR images with clinical and radiomic data.

Brain metastasis (BM) is one of the main complications of many cancers, and the most frequent malignancy of the central nervous system. Imaging studies of BMs are routinely used for diagnosis of disease, treatment planning and follow-up. Artificial Intelligence (AI) has great potential to provide automated tools to assist in the management of disease. However, AI methods require large datasets for training and validation, and to date there have been just one publicly available imaging dataset of 156 BMs. This paper publishes 637 high-resolution imaging studies of 75 patients harboring 260 BM lesions, and their respective clinical data. It also includes semi-automatic segmentations of 593 BMs, including pre- and post-treatment T1-weighted cases, and a set of morphological and radiomic features for the cases segmented. This data-sharing initiative is expected to enable research into and performance evaluation of automatic BM detection, lesion segmentation, disease status evaluation and treatment planning methods for BMs, as well as the development and validation of predictive and prognostic tools with clinical applicability.

Metabolic activity grows in human cancers pushed by phenotypic variability.

Different evolutionary processes push cancers to increasingly aggressive behaviors, energetically sustained by metabolic reprogramming. The collective signature emerging from this transition is macroscopically displayed by positron emission tomography (PET). In fact, the most readily PET measure, the maximum standardized uptake value (SUVmax), has been found to have prognostic value in different cancers. However, few works have linked the properties of this metabolic hotspot to cancer evolutionary dynamics. Here, by analyzing diagnostic PET images from 512 patients with cancer, we found that SUVmax scales superlinearly with the mean metabolic activity (SUVmean), reflecting a dynamic preferential accumulation of activity on the hotspot. Additionally, SUVmax increased with metabolic tumor volume (MTV) following a power law. The behavior from the patients data was accurately captured by a mechanistic evolutionary dynamics model of tumor growth accounting for phenotypic transitions. This suggests that non-genetic changes may suffice to fuel the observed sustained increases in tumor metabolic activity.

Growth dynamics of brain metastases differentiate radiation necrosis from recurrence.

Background: Radiation necrosis (RN) is a frequent adverse event after fractionated stereotactic radiotherapy (FSRT) or single-session stereotactic radiosurgery (SRS) treatment of brain metastases (BMs). It is difficult to distinguish RN from progressive disease (PD) due to their similarities in the magnetic resonance images. Previous theoretical studies have hypothesized that RN could have faster, although transient, growth dynamics after FSRT/SRS, but no study has proven that hypothesis using patient data. Thus, we hypothesized that lesion size time dynamics obtained from growth laws fitted with data from sequential volumetric measurements on magnetic resonance images may help in discriminating recurrent BMs from RN events. Methods: A total of 101 BMs from different institutions, growing after FSRT/SRS (60 PDs and 41 RNs) in 86 patients, displaying growth for at least 3 consecutive MRI follow-ups were selected for the study from a database of 1031 BMs. The 3 parameters of the Von Bertalanffy growth law were determined for each BM and used to discriminate statistically PDs from RNs. Results: Growth exponents in patients with RNs were found to be substantially larger than those of PD, due to the faster, although transient, dynamics of inflammatory processes. Statistically significant differences (P < .001) were found between both groups. The receiver operating characteristic curve (AUC = 0.76) supported the ability of the growth law exponent to classify the events. Conclusions: Growth law exponents obtained from sequential longitudinal magnetic resonance images after FSRT/SRS can be used as a complementary tool in the differential diagnosis between RN and PD.

CoQ10 reduces glioblastoma growth and infiltration through proteome remodeling and inhibition of angiogenesis and inflammation.

PURPOSE: Most monotherapies available against glioblastoma multiforme (GBM) target individual hallmarks of this aggressive brain tumor with minimal success. In this article, we propose a therapeutic strategy using coenzyme Q10 (CoQ10) as a pleiotropic factor that crosses the blood-brain barrier and accumulates in cell membranes acting as an antioxidant, and in mitochondrial membranes as a regulator of cell bioenergetics and gene expression. METHODS: Xenografts of U251 cells in nu/nu mice were used to assay tumor growth, hypoxia, angiogenesis, and inflammation. An orthotopic model was used to explore microglial infiltration, tumor growth, and invasion into the brain parenchyma. Cell proliferation, migration, invasion, proteome remodeling, and secretome were assayed in vitro. Conditioned media were used to assay angiogenesis, monocyte chemoattraction, and differentiation into macrophages in vitro. RESULTS: CoQ10 treatment decreased tumor volume in xenografts and orthotopic models, although its effect on tumor cell proliferation was not direct. Tumors from mice treated with CoQ10 were less hypoxic and vascularized, having less infiltration from inflammatory cells. Treatment-induced downregulation of HIF-1α and NF-kB led to a complete remodeling of the tumor cells proteome and secretome, impacting angiogenesis, monocyte infiltration, and their differentiation into macrophages. Besides, tumor cell migration and invasion were drastically restricted by mechanisms involving modulation of the actin cytoskeleton and downregulation of matrix metalloproteases (MMPs). CONCLUSIONS: CoQ10 has a pleiotropic effect on GBM growth, targeting several hallmarks simultaneously. Thus, its integration into current treatments of this fatal disease should be considered.

On optimal temozolomide scheduling for slowly growing glioblastomas.

Background: Temozolomide (TMZ) is an oral alkylating agent active against gliomas with a favorable toxicity profile. It is part of the standard of care in the management of glioblastoma (GBM), and is commonly used in low-grade gliomas (LGG). In-silico mathematical models can potentially be used to personalize treatments and to accelerate the discovery of optimal drug delivery schemes. Methods: Agent-based mathematical models fed with either mouse or patient data were developed for the in-silico studies. The experimental test beds used to confirm the results were: mouse glioma models obtained by retroviral expression of EGFR-wt/EGFR-vIII in primary progenitors from p16/p19 ko mice and grown in-vitro and in-vivo in orthotopic allografts, and human GBM U251 cells immobilized in alginate microfibers. The patient data used to parametrize the model were obtained from the TCGA/TCIA databases and the TOG clinical study. Results: Slow-growth "virtual" murine GBMs benefited from increasing TMZ dose separation in-silico. In line with the simulation results, improved survival, reduced toxicity, lower expression of resistance factors, and reduction of the tumor mesenchymal component were observed in experimental models subject to long-cycle treatment, particularly in slowly growing tumors. Tissue analysis after long-cycle TMZ treatments revealed epigenetically driven changes in tumor phenotype, which could explain the reduction in GBM growth speed. In-silico trials provided support for implementation methods in human patients. Conclusions: In-silico simulations, in-vitro and in-vivo studies show that TMZ administration schedules with increased time between doses may reduce toxicity, delay the appearance of resistances and lead to survival benefits mediated by changes in the tumor phenotype in slowly-growing GBMs.

Patient-specific forecasting of postradiotherapy prostate-specific antigen kinetics enables early prediction of biochemical relapse.

The detection of prostate cancer recurrence after external beam radiotherapy relies on the measurement of a sustained rise of serum prostate-specific antigen (PSA). However, this biochemical relapse may take years to occur, thereby delaying the delivery of a secondary treatment to patients with recurring tumors. To address this issue, we propose to use patient-specific forecasts of PSA dynamics to predict biochemical relapse earlier. Our forecasts are based on a mechanistic model of prostate cancer response to external beam radiotherapy, which is fit to patient-specific PSA data collected during standard posttreatment monitoring. Our results show a remarkable performance of our model in recapitulating the observed changes in PSA and yielding short-term predictions over approximately 1 year (cohort median root mean squared error of 0.10-0.47 ng/mL and 0.13 to 1.39 ng/mL, respectively). Additionally, we identify 3 model-based biomarkers that enable accurate identification of biochemical relapse (area under the receiver operating characteristic curve > 0.80) significantly earlier than standard practice (p < 0.01).

A Head-to-Head Comparison of 18F-Fluorocholine PET/CT and Conventional MRI as Predictors of Outcome in IDH Wild-Type High-Grade Gliomas.

(1) Aim: To study the associations between imaging parameters derived from contrast-enhanced MRI (CE-MRI) and 18F-fluorocholine PET/CT and their performance as prognostic predictors in isocitrate dehydrogenase wild-type (IDH-wt) high-grade gliomas. (2) Methods: A prospective, multicenter study (FuMeGA: Functional and Metabolic Glioma Analysis) including patients with baseline CE-MRI and 18F-fluorocholine PET/CT and IDH wild-type high-grade gliomas. Clinical variables such as performance status, extent of surgery and adjuvant treatments (Stupp protocol vs others) were obtained and used to discriminate overall survival (OS) and progression-free survival (PFS) as end points. Multilesionality was assessed on the visual analysis of PET/CT and CE-MRI images. After tumor segmentation, standardized uptake value (SUV)-based variables for PET/CT and volume-based and geometrical variables for PET/CT and CE-MRI were calculated. The relationships among imaging techniques variables and their association with prognosis were evaluated using Pearson's chi-square test and the t-test. Receiver operator characteristic, Kaplan−Meier and Cox regression were used for the survival analysis. (3) Results: 54 patients were assessed. The median PFS and OS were 5 and 11 months, respectively. Significant strong relationships between volume-dependent variables obtained from PET/CT and CE-MRI were found (r > 0.750, p < 0.05). For OS, significant associations were found with SUVmax, SUVpeak, SUVmean and sphericity (HR: 1.17, p = 0.035; HR: 1.24, p = 0.042; HR: 1.62, p = 0.040 and HR: 0.8, p = 0.022, respectively). Among clinical variables, only Stupp protocol and age showed significant associations with OS and PFS. No CE-MRI derived variables showed significant association with prognosis. In multivariate analysis, age (HR: 1.04, p = 0.002), Stupp protocol (HR: 2.81, p = 0.001), multilesionality (HR: 2.20, p = 0.013) and sphericity (HR: 0.79, p = 0.027) derived from PET/CT showed independent associations with OS. For PFS, only age (HR: 1.03, p = 0.021) and treatment protocol (HR: 2.20, p = 0.008) were significant predictors. (4) Conclusions: 18F-fluorocholine PET/CT metabolic and radiomic variables were robust prognostic predictors in patients with IDH-wt high-grade gliomas, outperforming CE-MRI derived variables.

SUVmax to tumor perimeter distance: a robust radiomics prognostic biomarker in resectable non-small cell lung cancer patients.

OBJECTIVE: The purpose of this study was to evaluate the prognostic value of novel geometric variables obtained from pre-treatment [18F]FDG PET/CT with respect to classical ones in patients with non-small cell lung cancer (NSCLC). METHODS: Retrospective study including stage I-III NSCLC patients with baseline [18F]FDG PET/CT. Clinical, histopathologic, and metabolic parameters were obtained. After tumor segmentation, SUV and volume-based variables, global texture, sphericity, and two novel parameters, normalized SUVpeak to centroid distance (nSCD) and normalized SUVmax to perimeter distance (nSPD), were obtained. Early recurrence (ER) and short-term mortality (STM) were used as end points. Univariate logistic regression and multivariate logistic regression with respect to ER and STM were performed. RESULTS: A cohort of 173 patients was selected. ER was detected in 49/104 of patients with recurrent disease. Additionally, 100 patients died and 53 had STM. Age, pathologic lymphovascular invasion, lymph nodal infiltration, TNM stage, nSCD, and nSPD were associated with ER, although only age (aOR = 1.06, p = 0.002), pathologic lymphovascular invasion (aOR = 3.40, p = 0.022), and nSPD (aOR = 0.02, p = 0.018) were significant independent predictors of ER in multivariate analysis. Age, lymph nodal infiltration, TNM stage, nSCD, and nSPD were predictors of STM. Age (aOR = 1.05, p = 0.006), lymph nodal infiltration (aOR = 2.72, p = 0.005), and nSPD (aOR = 0.03, p = 0.022) were significantly associated with STM in multivariate analysis. Coefficient of variation (COV) and SUVmean/SUVmax ratio did not show significant predictive value with respect to ER or STM. CONCLUSION: The geometric variables, nSCD and nSPD, are robust biomarkers of the poorest outcome prediction of patients with NSCLC with respect to classical PET variables. KEY POINTS: • In NSCLC patients, it is crucial to find prognostic parameters since TNM system alone cannot explain the variation in lung cancer survival. • Age, lymphovascular invasion, lymph nodal infiltration, and metabolic geometrical parameters were useful as prognostic parameters. • The displacement grade of the highest point of metabolic activity towards the periphery assessed by geometric variables obtained from [18F]FDG PET/CT was a robust biomarker of the poorest outcome prediction of patients with NSCLC.

Optimal Combinations of Chemotherapy and Radiotherapy in Low-Grade Gliomas: A Mathematical Approach.

Low-grade gliomas (LGGs) are brain tumors characterized by their slow growth and infiltrative nature. Treatment options for these tumors are surgery, radiation therapy and chemotherapy. The optimal use of radiation therapy and chemotherapy is still under study. In this paper, we construct a mathematical model of LGG response to combinations of chemotherapy, specifically to the alkylating agent temozolomide and radiation therapy. Patient-specific parameters were obtained from longitudinal imaging data of the response of real LGG patients. Computer simulations showed that concurrent cycles of radiation therapy and temozolomide could provide the best therapeutic efficacy in-silico for the patients included in the study. The patient cohort was extended computationally to a set of 3000 virtual patients. This virtual cohort was subject to an in-silico trial in which matching the doses of radiotherapy to those of temozolomide in the first five days of each cycle improved overall survival over concomitant radio-chemotherapy according to RTOG 0424. Thus, the proposed treatment schedule could be investigated in a clinical setting to improve combination treatments in LGGs with substantial survival benefits.

A Mathematical Description of the Bone Marrow Dynamics during CAR T-Cell Therapy in B-Cell Childhood Acute Lymphoblastic Leukemia.

Chimeric Antigen Receptor (CAR) T-cell therapy has demonstrated high rates of response in recurrent B-cell Acute Lymphoblastic Leukemia in children and young adults. Despite this success, a fraction of patients' experience relapse after treatment. Relapse is often preceded by recovery of healthy B cells, which suggests loss or dysfunction of CAR T-cells in bone marrow. This site is harder to access, and thus is not monitored as frequently as peripheral blood. Understanding the interplay between B cells, leukemic cells, and CAR T-cells in bone marrow is paramount in ascertaining the causes of lack of response. In this paper, we put forward a mathematical model representing the interaction between constantly renewing B cells, CAR T-cells, and leukemic cells in the bone marrow. Our model accounts for the maturation dynamics of B cells and incorporates effector and memory CAR T-cells. The model provides a plausible description of the dynamics of the various cellular compartments in bone marrow after CAR T infusion. After exploration of the parameter space, we found that the dynamics of CAR T product and disease were independent of the dose injected, initial B-cell load, and leukemia burden. We also show theoretically the importance of CAR T product attributes in determining therapy outcome, and have studied a variety of possible response scenarios, including second dosage schemes. We conclude by setting out ideas for the refinement of the model.

Nonlinear science against the COVID-19 pandemic.

This special issue showcases recent uses of mathematical and nonlinear science methods in the study of different problems arising in the context of the COVID-19 pandemic. The sixteen original research papers included in this collection span a wide spectrum of studies including classical epidemiological models, new models accounting for COVID-19 specificities, non-pharmaceutical control measures, network models and other problems related to the pandemic.

Dynamical properties of feedback signalling in B lymphopoiesis: A mathematical modelling approach.

Haematopoiesis is the process of generation of blood cells. Lymphopoiesis generates lymphocytes, the cells in charge of the adaptive immune response. Disruptions of this process are associated with diseases like leukaemia, which is especially incident in children. The characteristics of self-regulation of this process make them suitable for a mathematical study. In this paper we develop mathematical models of lymphopoiesis using currently available data. We do this by drawing inspiration from existing structured models of cell lineage development and integrating them with paediatric bone marrow data, with special focus on regulatory mechanisms. A formal analysis of the models is carried out, giving steady states and their stability conditions. We use this analysis to obtain biologically relevant regions of the parameter space and to understand the dynamical behaviour of B-cell renovation. Finally, we use numerical simulations to obtain further insight into the influence of proliferation and maturation rates on the reconstitution of the cells in the B line. We conclude that a model including feedback regulation of cell proliferation represents a biologically plausible depiction for B-cell reconstitution in bone marrow. Research into haematological disorders could benefit from a precise dynamical description of B lymphopoiesis.