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Resume La technique de reconstruction osseuse decrite par Masquelet, combinant l’induction d’une membrane in situ et une autogreffe osseuse spongieuse, permet d’envisager la reconstruction de pertes de substance osseuse diaphysaires etendues. Actuellement, cette technique est essentiellement limitee par la quantite de tissu osseux disponible chez les patients et par la morbidite induite par des prelevements massifs. Le but de cette etude etait d’etudier le comportement d’un implant cylindrique de ceramique macroporeuse, traite par un facteur de croissance osseuse et protege par une membrane induite en site heterotopique, susceptible de se substituer a une autogreffe osseuse. Materiels et methodes Soixante implants de phosphate d’hydroxyapatite tricalcique, sous forme de granules, dont 40 avaient ete traites par un facteur de croissance osseux (rhOP-1) ont ete mis en place dans des tunnels sous-cutanes ou au sein de membranes prealablement induites sur le dos de lapins. Un groupe d’implants etait examine a quatre semaines et l’autre a 16 semaines. Trois niveaux de coupe etaient examines pour chaque implant (extremites et milieu). Resultats Aucun developpement de tissu osseux n’a ete observe sur les implants du groupe non traite par le facteur de croissance osseux. Les implants mis en place au sein d’une membrane induite presentaient moins de resorption. La presence de tissu osteoide semblait correlee de facon significative a la mise en place de l’implant au sein d’une membrane induite a tous les temps de l’etude. Conclusion Le role de protecteur de la membrane induite semble confirme, ainsi que son role potentiel dans le phenomene d’osteogenese.

Resume Situe a l'interface entre la recherche fondamentale et la recherche clinique, le CIT biomateriaux de Bordeaux associe les industriels a l'environnement hospitalo-universitaire afin de valoriser la recherche biomedicale dans le domaine des biomateriaux. Ainsi, le CIT de Bordeaux apporte un soutien methodologique dans les phases de conception, developpement, evaluation que les projets soient en amont du marquage CE, concernent le marquage CE ou soient en aval du marquage CE. Les trois grands axes d'activite du CIT biomateriaux sont la recherche preclinique in vitro, la recherche preclinique in vivo et la recherche clinique (etudes cliniques pilotes). « L'etude de la modification du corail par traitements thermiques et fluides supercritiques en vue de son amelioration en temps que substitut osseux en odontologie et orthopedie » est un projet type encadre par le CIT de Bordeaux. L'objectif de ce projet etait d'evaluer l'efficacite de differentes techniques d'extraction matricielle innovantes et leur repercussion sur la biocompatibilite et les performances mecaniques du corail, afin d'obtenir un substitut osseux naturel entierement mineral.

To study the influence of smooth muscle cells (SMC) on endothelial cells (EC), different co-culture designs are available, including EC seeding on SMC extracellular matrix (ECM). We explored human umbilical vein endothelial cell (HUVEC) adhesion and proliferation on either in situ or coated ECM, elaborated by HUVECs or human arterial smooth muscle cells (HUASMCs), in the presence of different nutrient media containing varying amounts of fetal calf serum. Coating wells with HUVEC or HUASMC ECMs did not improve HUVEC adhesion 1 h after cell seeding, compared with uncoated wells. HUVEC adhesion on in situ HUVEC-ECM and HUASMC-ECM was significantly increased compared with uncoated wells. The substratum upon which cells are maintained was found to play a crucial role, in conjunction with the medium to which HUVECs are exposed for their proliferative response. These results stress the importance of selecting media in relation to the particular substratum, in order to avoid misinterpretation of data.

Research in biomaterials for bone reconstruction has led to elaborate osteogenic composites that combine porous ceramics with bone marrow stromal cells. The aim of this study was to evaluate the influence of direct vascularization of such composites on osteogenesis and the ability to produce a vascularized bone substitute transplant in an ectopic muscular site. Sixty-four coralline biomaterials were implanted in 32 Fisher rats under four conditions: (1) alone (reference group M, n = 16), (2) coated with bone marrow stromal cells (group MC, n = 16), (3) combined with a vascular pedicle (group MV, n = 16), or (4) coated with bone marrow stromal cells and combined with a vascular pedicle (MCV group, n = 16). The number of vessels in the pores (vessel-pore ratio) of the implants and the proportion of pores showing bone ingrowth (bone-pore ratio) were measured at 2, 4, 6, and 8 weeks on four implants of each group. Compared with the reference group, angiogenesis was higher when the biomaterial was combined with a vascular pedicle or was coated with osteoprogenitor cells. The association of both vascular pedicle and osteoprogenitor cells increased vascularization by 60 percent (p = 0.003) and osteogenesis by 62 percent (p < 0.001). A combination of both vascular pedicle and bone marrow osteoprogenitor cells in coralline implants enhances neovascularization and osteogenesis after implantation in ectopic intramuscular sites to a greater extent than either does alone.

B-type natriuretic peptide (BNP) is a hormone mainly secreted by cardiac ventricle myocytes and which is increased in cardiac diseases. Moreover, BNP expression has been shown in various cell/tissue types. Six different human endothelial cell (EC) culture models arising from macro and microcirculation either primary cultures or cell lines were cultured and screened for BNP presence and secretion. All cell types expressed BNP mRNA while only the ECs arising from bone marrow stromal compartment secreted high amounts of BNP protein. This report is the first to identify ECs as a new source of BNP. However, BNP secretion is limited to a particular EC type.

Bone development and remodeling depend on complex interactions between bone-forming osteoblasts and other cells present within the bone microenvironment, particularly vascular endothelial cells that may be pivotal members of a complex interactive communication network in bone. Our aim was to investigate the interaction between human umbilical vein endothelial cells (HUVEC) and human bone marrow stromal cells (HBMSC). Cell differentiation analysis performed with different cell culture models revealed that alkaline phosphatase activity and type I collagen synthesis were increased only by the direct contact of HUVEC with HBMSC. This “juxtacrine signaling” could involve a number of different heterotypic connexions that require adhesion molecules or gap junctions. A dye coupling assay with Lucifer yellow demonstrated a functional coupling between HUVEC and HBMSC. Immunocytochemistry revealed that connexin43 (Cx43), a specific gap junction protein, is expressed not only in HBMSC but also in the endothelial cell network and that these two cell types can communicate via a gap junctional channel constituted at least by Cx43. Moreover, functional inhibition of the gap junction by 18α-glycyrrhetinic acid treatment or inhibition of Cx43 synthesis with oligodeoxyribonucleotide antisense decreased the effect of HUVEC cocultures on HBMSC differentiation. This stimulation could be mediated by the intercellular diffusion of signaling molecules that permeate the junctional channel.

The attachment of human umbilical vein endothelial cells (HUVECs) on substrates that had been covalently grafted with the cell adhesion peptides Arg-Gly-Asp (RGD) was investigated. This approach was used to provide substrates that are adhesive to cells even in the absence of serum proteins and to cells that have had no prior treatment of the surface with proteins that promote cell adhesion. We wanted to improve control of cellular interactions with cell-adhesive materials by providing fixedly bound adhesion ligands. Silica was examined as a model surface. The peptides were grafted using three different steps: grafting of aminosilane molecules; reaction with a maleimide molecule; and immobilization of cell-binding peptides containing the RGD sequence. The RGD-grafted surface was characterized by X-ray photoelectron spectroscopy (XPS) and contact-angle measurements.

To select an insert suitable for human umbilical vein endothelial cell (HUVEC) culture, we compared several available inserts of 0.2 to 0.45 micron porosity: Cellagen (ICN), Transwell-COL (Costar), Millicell-HA and CM (Millipore), Anopore (Nunc), Cyclopore (Falcon) in comparison with a control surface (Thermanox). The requirements were: (i) to promote attachment, adhesion and proliferation of HUVEC (judged by [3H]thymidine incorporation into DNA at days 1, 3, 7); (ii) to allow HUVEC visualization by inverted, fluorescence microscopy for uptake of DiI-Ac-LDL and scanning electron microscopy, performed at day 9 after seeding. Because Transwell and Cellagen are collagen precoated and CM has to be coated for cell culture, we performed collagen coating (types I + III or IV) for non-pretreated inserts for the purpose of comparison. Our preferences comprise Transwell-COL, Cyclopore not coated or coated (whatever the collagen type), and Cellagen. However, on a quality/price ratio criterion, Cyclopore, even uncoated, is the insert of choice. The HA, CM and Anopore inserts, even coated, do not allow HUVEC growth but do not alter positive uptake of acetylated LDL.

Based on a new concept, a procedure combining induced membranes and cancellous autografts allows the reconstruction of wide diaphyseal defects. To date, this procedure is limited by the amount of cancellous bone available from the patient and by the related morbidity at the donor site. The aim of this study was to evaluate the biological effect of induced membranes on a cylindrical-shaped ceramic implants loaded with OP-1 in heterotopic site.Sixty hydroxyapatite tricalcium phosphate (HA-TCP) implants, 20 of which being loaded with a bone growth factor (rhOP-1) were inserted either in a subcutaneous tunnel or within a previously induced membrane on the back of rabbits. There were two time-points at four and 16 weeks. Implants were investigated at three different levels (extremities and middle).None of the untreated implants showed any evidence of bone formation. Implants inserted in an induced membrane presented with less resorption. Bone ingrowth within the pores of the materials was significantly higher when the implants were inserted into the induced membrane whatever the time-point considered.The membrane seems to play the role it was assigned, i.e. to protect and revascularize the implant, thus favouring osteogenesis that occurs in 80% of the implants after four months.

The concept of endothelial cell seeding of vascular prostheses is designed as a method for improving long-term patency substitutes because endothelium is considered as the haemocompatible surface of reference. The assessment of the functionality of cells lining a biomaterial is thus of crucial importance. We have reported encouraging results concerning the ability of a polyester coated with albumin and chitosan (M 11) to be lined by a confluent monolayer of cultured human endothelial cells (EC). The aim of the present work was to study the expression of thrombomodulin (membrane glycoprotein responsible for anticoagulant activity) in EC lying on M.11 by anticoagulant activity and mRNA level with and without stimulation. Results obtained in basal conditions showed that EC on M.11 have a comparable expression of TM mRNA when compared to control (EC on tissue culture plates) despite a lower TM surface activity for EC on the biomaterial. In terms of ratio (stimulated cells to unstimulated ones) the response in activity for EC on M.11 is comparable to that of the control. These results indicate that cells lying on M.11 are able to respond to physiological-like stimuli, despite a tendency for these cells to express a procoagulant phenotype when compared with control EC.

Angiogenesis is a tightly regulated process involved in growth, repair, and bone remodeling. Several studies have shown that there is a reciprocal regulation and functional relationship between endothelial cells and osteoblast-like cells during osteogenesis, where systemic hormones and paracrine growth factors play an active role. Angiogenesis is induced by a variety of growth factors; among them vascular endothelial growth factor (VEGF) may be an important mediator for the angiogenic process involved in bone physiology. We studied the VEGF effect on osteoblast progenitor cells (Human Bone Marrow Stromal Cells: HBMSE) cultured alone or associated with endothelial cells (Human Umbilical Vein Endothelial Cells: HUVEC) in different co-culture models (co-culture with or without direct contact, conditioned medium), to determine the influence of VEGF on these cells and on their relationship. In agreement with other studies, we show that HBMSC express and synthesize VEGF, HUVEC conditioned medium has a proliferative effect on them, and early osteoblastic marker (Alkaline phosphatase activity) levels increase when these cells are co-cultured with HUVEC only in direct contact. However, unlike previous studies, we did not find that VEGF increased these processes. These results suggest that the intercommunication between endothelial cells and osteoblastic-like cells requires not only diffusible factors, but also involving cell membrane proteins.

The attachment of human umbilical vein endothelial cells (HUVECs) on substrates that had been covalently grafted with the cell adhesion peptides Arg-Gly-Asp (RGD) was investigated. This approach was used to provide substrates that are adhesive to cells even in the absence of serum proteins and to cells that have had no prior treatment of the surface with proteins that promote cell adhesion. We wanted to improve control of cellular interactions with cell-adhesive materials by providing fixedly bound adhesion ligands. Silica was examined as a model surface. The peptides were grafted using three different steps: grafting of aminosilane molecules; reaction with a maleimide molecule; and immobilization of cell-binding peptides containing the RGD sequence. The RGD-grafted surface was characterized by X-ray photoelectron spectroscopy (XPS) and contact-angle measurements.

The paper deals with the in vitro assessment of endothelial cell (EC) phenotype covering an albumin- and chitosan-coated polyester fabric and shows that resting ECs express a pro-coagulant phenotype by releasing a high von Willebrand factor level and expressing low thrombomodulin surface activity, despite maintaining an adequate response to stimulating agents.

Ru/MoO 3 catalysts containing 0.5, 1 and 2 wt% Ru have been prepared using two kinds of MoO 3 , and their reduction by H 2 has been followed by XRD and TGA. Ru catalyses the reduction of the MoO 3 ; a hydrogen Mo bronze is first formed, and reduction to Mo metal is complete at 773 K. Catalysts reduced at 623 K or above give ∼ 25-30% of C 1 -C 3 alcohols from syngas; the proportion of C 2 or C 3 alcohols decreases with reduction temperature, and when this is 723 K only methanol is formed. Reduced MoO x species promote alcohol synthesis either through an SMSI effect or through a dual-site mechanism.

Adiabatic large amplitude motion along a path of minimal potential energy is shown to be coupled to residual modes of excitation. These are described by RPA modes denned at every point q of the adiabatic path. A semi-classical Hamiltonian for this system is constructed, which exhibits the coupling terms between the two types of modes.

In this paper it is demonstrated that there is a fruitful analogy between nuclear and molecular collective rotations and vibrations. In order to implement this analogy, and to construct the analog of the factorable molecular Born-Oppenheimer trial function, one must transform into a representa- tion, in which the nuclear collective “position” operators and the residual intrinsic operators are diagonal. In this representation, the trial function is assumed factorable. Using projection operator techniques, this trial function can be transformed back into a particle coordinate representation, so that conventional many-body techniques can be used. The collective operators are “chosen” only to the extent needed to identify the “type” of collective process that one intends to describe, by stating their transformation properties under the standard symmetry operations in coordinate-, spin- and isospin space: In the present case, the collective operators form a symmetric, second rank tensor, are isoscalar, and thus describe rotations, and size and quadrupole shape oscillations. A more detailed determination of these operators is based on the dynamical argument of minimal coupling between collective and intrinsic modes of excitation. One then finds that in the harmonic limit, vibrations have frequencies given by RPA type equations, and rotational moments of inertia are determined by self-consistent cranking type prescriptions.

The equations of motion for a system of interacting fermions with SU(3) symmetry are established in TDHF and ATDHF approximations. These equations are presented in a form such that the comparison between different approximations can be made in an analytic way. The problem of coupling between large and small amplitude modes is also discussed.

This paper attempts to implement the analogy between nuclear collective dynamics and the dynamics of molecular rotation and vibration. The procedure used is illustrated for the case of a single canonical pair Q , K of collective operators. It contains two essential steps. First, the construction of a factorable trial function modelled after the well-known Born-Oppenheimer ansatz. Redundant degrees of freedom are eliminated by projection. Second, a decomposition of the hamiltonian into a series, each term being the product of a power of the collective momentum K and of a cofactor which commutes with the collective position operator Q . This decomposition is unique. These two steps, used in conjunction, lead to an expression for the mean energy in terms of a collective hamiltonian H (p, q) . Two optimization conditions are applied, which jointly determine both the intrinsic wave function and the exact form of the collective operator Q (given its general nature). As a result of several well-defined approximations in the evaluation of H (p, q) , a self-consistent cranking-type expression energes for the inertial parameter. Small amplitude oscillations near the unconstrained equilibrium are described by RPA-type equations.

This work deals both with the construction of a new analytically solvable model and with the quantitative test of the time-dependent Hartree-Fock (TDHF) method. First, we construct a new analytically solvable model, which serves as a testing ground for the various many-body approximation methods. The construction is based on two vector operators that are the generators of a Lie algebra. The model consists of a one-dimensional system of two distinguishable sets of fermions interacting via a schematic two-body force. The model has a simple analytic energy spectrum. Second, we use this model to test the validity of the TDHF approximation. Exact eigenvalues are compared with the corresponding solutions of the TDHF method. The TDHF approximation is shown to be reasonably accurate in the description of the system's eigenstates.

The adiabatic limit of time-dependent Hartree-Fock theory (ATDHF) is developed in a systematic way so as to provide a basis for the description of large amplitude, slow, collective motion in nuclei. A variational principle is used to find the best approximation to a solution of the time-dependent Hartree-Fock equation in the form of a parametrized wave packet φ(p, q), p, q being time-dependent c-numbers. The mean energy for the wave packet φ(p, q) defines a classical collective Hamiltonian H(p, q). The evolution of the parameters p and q is described by the canonical equations of motion associated with H. A set of self-consistency conditions, of a formally stationary type, determines possible optimal wave packets. An iteration scheme for the solution of the self-consistency conditions is proposed. The quantal interpretation of the classical description of collective motion embodied in the Hamiltonian H(p, q) is discussed. Finally, two simple, well understood illustrative examples are presented to illustrate the application of the general formalism developed in this paper.

The validity of the adiabatic nuclear Born-Oppenheimer (NBO) approximation method is investigated by means of an analytically solvable model. The NBO equation of collective motion derived, when this method is applied to the model, is shown to have the structure of a Schrodinger equation. The NBO energy spectrum is then obtained by numerical integration of this equation and compared with the analytic energy spectrum. We show that the NBO approximation is very accurate in the description of the system's eigenstates. The time-dependent Hartree-Fock (TDHF) results, obtained in a previous publication for the solvable model, are compared with their NBO counterparts. We find that, although both methods describe the system's states very well, the NBO approximation is more accurate in the adiabatic domain.

It is well known that the existence of charge carrying nuclear fields, resulting in an exchange of charge between nucleons, also gives rise to additional electromagnetic interaction terms for the nuclear system. It is commonly assumed that these effects are zero for the deuteron ground state, on account of the symmetry properties of the exchange current. The present investigation aims to draw attention to the fact that this conclusion is only true in the adiabatic approximation, which neglects the recoil of the nucleons. In the expression for the electromagnetic interaction of charge exchanging nucleons in motion, nuclear "two-particle currents" appear, in addition to the meson currents encountered in the adiabatic approximation. Whereas the latter are antisymmetric under charge exchange of a nucleon pair, the former are not, and give rise to electromagnetic exchange effects for the deuteron ground state.

It has been suggested that auroral backscattering of radio waves in the frequency range 100–800 Mc/s is a consequence of the joint action of turbulence and the earth's magnetic field, which was assumed to produce small-scale elongated irregularities of ionization lined up with the earth's field. In the present work we investigate solutions of the equation for diffusion and transport of ionization in a weakly ionized turbulent plasma in the presence of a magnetic ffeld. Because of space charge effects, the problem is nonlinear in the ionization density n(x, t). We show that in the linearized approximation the effect of the magnetic field on turbulence does not lead to irregularities of a scale sufficiently small to explain the observed data. Nonlinear effects may play a role and are being investigated. Preliminary results show that they are indeed capable of increasing density gradients in irregularities of sufficient amplitude.

A unified picture of the relativistic single particle wave mechanics for both the spin 0 boson and the spin 1/2 h fermion is presented. The single particle equations discussed are the Klein-Gordon equation for spin 0, and the Dirac equation for spin 1/2 h. The wave mechanics of these particles is discussed. The field theoretic interpretation, in which the wave equations give the equations of motion for field operators, is connected with the particle interpretation. (M.H.R.)

Determinantal many-particle wavefunctions, as used in Hartree-Fock theory of finite nuclei, are not eigenfunctions of the total linear momentum, nor, generally, of the total angular momentum. A simple derivation is given for the expression for the energy in terms of momentum- or angular momentum projected determinants. A variational principle based on this expression is used to determine the collective energy (center of mass kinetic and rotational energy). This method gives the correct energy of center of mass motion, and in the case of strong deformation, the Thouless-Valatin self-consistent cranking result for the rotational energy.

Within the Dyson equation approach to higher correlation functions we derive RPA-like equations for the 2-particle-1-hole problem. The equations can be entirely expressed by the RPA sub-solutions of the particle-hole and particle-particle problem. In this way our equations are very well suited for weak or intermediate coupling situations.

effects went almost parallel with the further development of the shell model. As a result, a synthesis of the two aspects of nuclear dynamics, single particle, and collective behaviour, has become one of today's most interesting problems of nuclear structure. Collective aspects had for a long time dominated the thinking in nuclear physics, ever since in 1937 Bohr and Kalckar presented the liquid drop

The nuclear Hamiltonian is expressed in terms of the total angular momentum variables, and of intrinsic particle coordinates by means of a contact transformation. The intrinsic structure is approximated by a Hartree-Fock solution, and the Coriolis coupling diagonalized by admitting particle-hole excitations. It is shown that in this approximation, the rotational energy takes the conventional form, as given by the “unified” model, the moment of inertia being determined by a self-consistent “cranking” formula. This, and corresponding results for matrix elements of multipole operators substantiate the validity of the more intuitive procedures based on the unified model and on the use of single particle orbitals in deformed potential wells.

A new, entirely quantal description of nuclear collective rotation is developed. The basic element is the explicit introduction of the pair (or pairs) of operators which describe the collective motion; in the case of rotation this is the total angular momentum J and a phase angle ϕ. A representation is then constructed, in which either J or ϕ is diagonal. The non-collective degrees of freedom are described in a subspace of the set of antisymmetrized independent particle wave functions φn; this subspace is obtained by using suitable projection operators. The formalism is developed for “nuclei” in two-dimensional space only. It is shown that the ϕ-representation is particularly useful and leads to the construction of an operator H J = H ( i ) + JH ( c ) + 1 2 J 2 Q , whose eigenvalues are the set of all levels with angular momentum J. These levels may be found by the usual HF and RPA procedures. It is shown that the moment of inertia of the ground state rotational band is given by the Thouless-Valatin formula.

This paper presents a theoretical analysis of the mechanism that enables transmission of vhf-signals over distances of the order of 103 km. It is found that turbulent mixing, operating at the lower edge of the E-layer (h=80-90 km) produces fluctuations in electron density of sufficient intensity to account for the observed signals. The basic assumptions are the existence of a sufficiently strong gradient of electron density (dN/dh > 103 cm-3/km) and a reasonable level of turbulent activity.

The linked-cluster expansion for the transition amplitude for deuteron stripping reactions is derived. It is shown that this amplitude can be expressed (exactly) as a matrix element of an effective neutron-proton interaction between appropriate two-nucleon states. A linked-cluster expansion of this effective interaction is given. The initial state is a deuteron scattering state in an effective potential, for which a linked-cluster expansion has been given previously. The final state is the product of a bound Hartree-Fock state for the stripped nucleon and a continuum state in an effective optical potential for the outgoing nucleon. The leading terms in this expression for the transition amplitude are compared with conventionally used approximate expressions for the stripping reaction (such as the direct interaction approximation) and certain questions, such as the theoretical interpretation of the spectroscopic factor and the single particle energy of the captured nucleon, are examined.

A systematic approach is made to the problem of determining the parameters of collective motion in nuclei. Formulas are given for the moment of inertia of the rotational energy and for the collective g -factor of the magnetic moment. These expressions are not based on the use of a model of the nucleus. Such a model will be needed, however, for the numerical evaluation of the expressions derived. The essential elements of a usable model are briefly discussed.

The theory of elastic nucleon-nucleus scattering is generalized to include the recoil of the nucleus. The exact T matrix is obtained as the residue of a second-order pole of an off-diagonal single particle Green function. A perturbation expansion is made using the intrinsic Hamiltonian and a new generalization of the Goldstone expansion for the ground state. This makes the extraction of the delta function for overall momentum conservation simple. A linked cluster expansion is derived for the Green function and two types of recoil corrections are identified, linked and unlinked. A new theorem is proved, expressing the exponent of a one body operator in a convenient way. This allows us to show that terms with linked corrections cannot contribute to the scattering. The T matrix including recoil is then expressed as the T matrix of the generalized optical potential calculated in the usual way (but using the intrinsic Hamiltonian) multiplied by a correction factor due to recoil. An approximation scheme for calculating the correction factor is presented and suggestions are made for improving existing calculations.

It is shown that the kinetic energy T of any system of interacting particles may be written in a “canonical form”, which displays the dependence of T on the total angular momentum. A formula is given for the zero order moments of inertia associated with collective rotation of the system. The formalism is applied to nuclei, and for purposes of comparison, to molecules. In addition, it is shown that an unsymmetric formulation of the many particle problem, in which only the nucleons of a “core” participate in the definition of the body fixed axes, leads to a Hamiltonian which is related by a simple averaging process with the “unified” model of Bohr and Mottelson. Finally, a connection is established between the “unified” model of Bohr-Mottelson and the “cranking model” of Inglis.

The formal method of regularization of mathematical expressions of sums of products of different types of $\ensuremath{\delta}$-functions is first applied to the example of vacuum polarization. It is emphasized that only a regularization of the whole expression without factorization leads to gauge invariant results. It is further shown, that for the regularization of the expression for the magnetic moment of the electron, a single auxiliary mass is sufficient, provided that different functions of the same particle (e.g., the photon functions $\overline{D}$ and ${D}^{(1)}$) are regularized in the same way and that the regularization of products of two electron functions is never factorized. The result is then the same as that of using Schwinger's method of introducing suitable parameters as new integration variables in the argument of $\ensuremath{\delta}$-functions, without using any auxiliary masses.