In this chapter we cover both polycythemia and fetomaternal hemorrhage. In the former, the hematocrit and blood volume are generally increased, whereas in the latter, both are generally very low. In both conditions, however, there is concern about inadequate organ blood flow, oxygen delivery, and potential for hypoxic injury, especially in the brain. A large body of definitive information on polycythemia is available that allows conclusions to be made as to management and patient outcome. However, the area of fetomaternal hemorrhage has not been as comprehensively investigated and relies mostly on small clinical series and case reports. There are no controlled basic or clinical trials, which hampers our understanding of pathophysiology as well as therapeutic modalities. This leaves us with gaps in our knowledge of this important topic. What is presented here is the most current body of information and recommendations for diagnosis, treatment, and reported outcomes.

1. Hyperleukocytosis may be the first presentation of acute myeloid leukemias and prognosticates high morbidity and mortality making immediate therapy imperative.

2. Neurologic deficits, respiratory failure with pulmonary infiltrates, fever and visual changes are frequent presentations of hyperleukocytosis due to leukocyte infiltrates or vessel occlusion. Antibiotics are warranted although the etiology may not be infectious.

3. Hyperleukocytosis can result in other oncologic emergencies such as leukostasis, tumor lysis syndrome, and/or disseminated intravascular coagulation involving multiple organ systems. Evaluation and work up should be directed by symptoms; consider evaluation for intracranial hemorrhage.

4. Differentiate between the true hyperkalemia of tumor lysis syndrome and pseudohyperkalemia of hyperleukocytosis with ECG and a heparinized plasma sample before treating.

5. Urgent cytoreduction with Hydroxyurea (Hydrea) and/or leukapheresis may need to be initiated in parallel to induction chemotherapy. If necessary, transfer to an institution that is able to provide these services immediately.

Mesh-Free FD Approximations are easy to work with and are in some sense optimal in their representations of functions locally around a single point. However, in more than one dimension, approximations based on polynomial interpolants encounter severe difficulties if the node points are not regularly placed (grid-based). Additional difficulties often arise at boundaries which (above 1-D) may be irregularly shaped, and also from mixtures of scales across a computational domain that may require spatially variable resolution. It transpires that radial basis functions (RBFs) can favorably replace (or supplement) polynomials in such situations. Their use in creating spatially local and highly effective FD-type approximations is quite recent.