1 Introduction

Medical imaging systems using the aspects of interactions of the non-ionizing rays with the body organs creating qualified images are able to improve diagnosis and treatment plan indirectly. MRI has been known as a powerful noninvasive tool on the medical field due to its unique features. Two quantities potentially in MRI indicating the important information on the objects are longitudinal (T1) and transverse (T2) relaxation times [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. A lot of the statistical and dynamical parameters can vary these times. For instance, they may be varied by the geometrical structural of the capillary bed at the different situations [20,21,22,23,24,25,26,27,28,29,30,31,32].

T1 is the unalterable progression of a spin system in the direction of thermal equilibrium with freedom orbital degrees of the medium into which the spins are inserted, the so-called lattice. This includes all observable spin quantities like spin–spin energies or various quantum coherences as well as transverse or longitudinal magnetizations. Only when evolution relies on the static spin–spin interactions has transverse relaxation or spin–spin relaxation been considered [33]. The types of tissues and the others have a unique T1-value resolving those in the image. The T2 relaxation raises from changing incrementally dephasing of rotating dipoles, which leads to a drop in magnetization in the transverse surface. This is done by tissue specific properties that mainly influence the motion ratio of the protons, which mostly originate from water molecules, and also by a changing local magnetic field while energy is transferred between dipoles oriented antiparallel and parallel to the external magnetic field and mutually rotate in opposite directions. This energy transfers or flip ratio between dipoles or spins augments when the frequency of the local magnetic field variation gets close to the Larmor frequency. It correlates with the translational and rotational ratio of the water molecule or neighboring dipoles. Meanwhile, the dipole–dipole communication is reinforced by the strength of the local field, which is due to the juxtaposition of neighboring dipoles. Since pure water molecules move significantly faster than the Larmor frequency, the T2 relaxation time is long, around 3 to 4 s. In pure water, the swift motion causes the T1 and T2 relaxation times to be roughly equal. The analytical relationship between these relaxation times has not been fully understood yet.

Combination of these times may obtain the new maps in which more information has been present towards better diagnosis in the medical field along with appropriate and greater signal to noise ratio (SNR) that has been addressed in this study. A parametric-function on these times is introduced based on the appropriate assumptions. This function is evaluated at the different values of the T1 and T2 on the various α-parameter values. Furthermore, the new images are obtained using the T1- and T2 -weighted maps based on the proposed function.

2 Materials and methods

After a magnetized spin system has been disturbed from its thermal equilibrium state by an RF pulse, it will return to this state, provided the external force is removed and sufficient time is given, according to the laws of thermodynamics. This process is characterized by a precession of \(\overrightarrow{\text{M}}\) about the B0 field, called free precession; a recovery of the longitudinal magnetization Mz, called longitudinal relaxation; and the destruction of the transverse magnetization Mxy, called transverse relaxation, as shown in Fig. 1.

Fig. 1
figure 1

The trajectory of the tip of \(\overrightarrow{\text{M}}\) during the relaxation as observed in the laboratory frame

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Both relaxation processes are often ascribed to the existence of time-dependent microscopic magnetic fields that surround a nucleus as a result of the random thermal motions present in an object. The relaxation process may be described using the Bloch equation. Phenomenologically, the transverse and longitudinal relaxations are described by a first-order process. Specifically, in the Larmor-rotating frame, one may obtain:

$$\frac{dM_{z'}}{dt}=-\frac{M_{z'}-M_z^0}{T_1},\;and\;\frac{dM_{x'y'}}{dt}=-\frac{M_{x'y'}}{T_2}$$
(1)

The time evolution for these components has been obtained as follow:

$$M_{x'y'}\left(t\right)=M_{x'y'}\left(0^+\right)e^{-{\textstyle\frac t{T_2}}},\;and\;M_{z'}\left(t\right)=M_z^0\left(1-e^{-{\textstyle\frac t{T_1}}}\right)+M_{z'}\left(0^+\right)e^{-{\textstyle\frac t{T_1}}}$$
(2)

where \({\text{M}}_{{x}{\prime}{y}{\prime}}\left({0}^{+}\right)\) and \({\text{M}}_{{z}{\prime}}\left({0}^{+}\right)\) are the magnetizations on the transverse plane and along the z-axis immediately after an RF pulse, and \({\text{M}}_{\text{z}}^{0}\) is, as before, the longitudinal magnetization at thermal equilibrium, as shown in Fig. 2. The values of T1 and T2 depend on the tissue composition, structure, and surroundings. For a given spin system, T1 is always longer than T2. In biological tissues, T1 is about 300 to 2000 ms, and T2 is about 30 to 150 ms [34].

Fig. 2
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Relaxation curves

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The amounts of magnetized vectors of longitudinal Mz (t) and transverse Mxy (t) are always related together as,

$$\frac{{T}_{1}}{{T}_{2}}=\frac{Ln\left[\frac{{M}_{xy}\left(t\right)}{{M}_{0}}\right]}{Ln\left[1-\frac{{M}_{z}\left(t\right)}{{M}_{0}}\right]}$$
(3)

To segment and extract some of information from the T1 and T2 maps together, a parametric function has been suggested between them as follows,

$${M}_{z}\left(\frac{{T}_{1}}{{T}_{2}}\right)={\left(\frac{{T}_{1}}{{T}_{2}}\right)}^{\alpha }\times {M}_{xy}\left[{\left(\frac{{T}_{1}}{{T}_{2}}\right)}^{\alpha }\right]$$
(4)

where α is the parameter. After substituting the equations of Mz (t) and Mxy (t) indicating in Fig. 2, into the Eq. (4), one may obtain the parametric function named transverse-longitudinal-function (TLF) as follows,

$$\mathrm{TLF}=\mathrm F\left({\mathrm T}_1,{\mathrm T}_2,\mathrm\alpha\right)=\mathrm T_1^{\mathrm\alpha}\times\exp\left(-\frac{\mathrm T_1^{\mathrm\alpha}}{\mathrm T_2^{\mathrm\alpha+1}}\right)+\mathrm T_2^{\mathrm\alpha}\times\exp\left(-\frac1{{\mathrm T}_2}\right)-\mathrm T_2^{\mathrm\alpha}=0$$
(5)

The TLF was simulated using Matlab (R2016a, MathWorks Inc., MA) software, and evaluated in terms of various relaxation times and the different amounts of α-parameter in order to obtain the new maps named TLFmax, αmin and αmax.

3 Results

The TLF was calculated in terms of various relaxation times [35] and the different amounts of α-parameter, as shown in Fig. 3 and Table 1.

Fig. 3
figure 3

The graphs of TLF via the α-parameter at the various values of relaxation times

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Table 1 The maximum values of TLF along with the correspondence α-parameter at the various T1 and T2 values in terms of ms [35]
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The behavior of TLF curve indicates to have a maximum point (TLFmax) at a distinct α-parameter (αmax), and then approaches to zero at the αmin-value. Additionally, Fig. 4 shows TLF behavior versus α-parameter for different T1 and T2 values to determine TLFmax besides αmax and αmin amounts. The maximum alpha value is exactly where the TLF value is maximized. Also, the minimum alpha value is where the TLF plot reaches zero.

Fig. 4
figure 4

Determining TLFmax, αmax and αmin amounts for distinct T1 and T2 values

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Due to the difference between the T1 and T2 values of some tissues, the T1 and T2 values are first calibrated according to the Table 1 and finally the derived image is normalized between zero and one. Therefore, the TLF function was applied to some experimental images with different alpha values. The values of TLFmax, αmin and αmax increase with increasing T1- and T2-values simultaneously, as shown in Fig. 5. The TLFmax also increases while the amounts of αmin and αmax increase. In our presented method, the SNR of TLFmax was larger than conventional T1 and T2 imaging methods. This advantage may be used to segment some of the tissues as well as to increase the SNR.

Fig. 5
figure 5

The variations of TLFmax, αmin and αmax in terms of the T1- and T2-values and related SNR

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For more investigation, the graphs of T2 in terms of T1 at various values of the α-parameter as 1, 1.5, 2, 2.5, 3, 10, 30, 50, 100, 150, 200, 250, 300, 350, 400 and 450 are calculated while the TLF approaches to zero, as shown in Fig. 6. The higher α-parameter value, the more linearity behavior of graph will be.

Fig. 6
figure 6

The graphs of T2 in terms of T1 at various values of the α-parameter at TLF = 0

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Figure 7 compares the T1, T2, T1/T2 weighted images and TLF function on experimental MR brain image for different α-parameter values. As it can be seen by applying the TLF function to the image, at low alpha values, the image is white with little details, and at alpha values higher than 2, the image becomes dark and becomes completely unclear.

Fig. 7
figure 7

The T1, T2, T1/T2 weighted images besides applying our proposed function of TLF to experimental MR brain image for different α-parameter values

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The seven images of T1- and T2-weighted indicated as numbers of #155, #417, #53, #45, #28, #86 and #81 from the references [36,37,38] were chosen for simulation and production of the maps of TLFmax, αmin and αmax, as shown in Fig. 8.

Fig. 8
figure 8figure 8figure 8figure 8

The T1/T2, TLFmax, αmin and αmax maps originated from seven T1- and T2-weighted images [36,37,38] based on the proposed function

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The values of SNR for the T1, T2, TLFmax, αmin and αmax maps are calculated [39], as shown in Figure 9. The SNR of the TLFmax is the most value in compared to the others.

Fig. 9
figure 9

The graph of SNR on the different maps for seven MR-images correspondence with Fig. 8

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4 Discussion

Relaxation times are the important parameters to characterize various objects as well as illustrate anatomy in MRI. With the receipt of the color-coded cards, the doctors received an extra diagnostic tool that expressively improves medical diagnostics [40]. Especially with the advance of resonance procedures, longitudinal relaxation has emerged as a dominant notion, both as a central subject in thermodynamics and as a tool of paramount importance in the study of structural and dynamic characteristics of condensed matter. With respect to the motivation of the relaxation times, we have considered them to obtain a parametric function as TLF in order to further investigation and production of new maps in which some features may be represented well. The results indicated that the new map has the good SNR. A theoretical combination of these relaxation times may be used to predict and segment some abnormalities as early stages.

Multiple factors influence the T1 relaxation time within the voxel of interest, such as substrate deposition and water contents. Elevated water content in the edema areas may have prolonged the T1 relaxation time [41]. In general, factors influencing the T1 and T2 relaxation times of dissimilar tissues are determined by interactions, size, and molecular motion. The protons that give rise to a nuclear MR signal are mostly found in lipids and cell water, although those in solids and proteins do not normally contribute to the obtained signal. This is because of the motion properties of molecules, which determine transverse and longitudinal relaxation times. The NMR process is a picture of the magnetized matter, with brightness signifying the degree of magnetization. The greater signal from a distinct tissue causes the larger transverse part of the magnetization. Enormous transverse magnetization leads to a great amplitude in the receiver coil, which lies in the transverse plane to acquire the image. The maps of TLFmax, αmin and αmax have indicated a hybrid combination of relaxation times that may mathematically help on the better diagnostics through gaining greater SNR.

A spinning proton can usually apply a magnetic field to its neighbors. And, the greatest regular source of lattice fields is the dipole–dipole interaction between adjacent spins. While the molecules are swirling through the space, the distance between the protons changes, resulting in a fluctuating magnetic field. If the field has some parts that adapt the Larmor frequency, protons in spin-down situation with higher-energy can return to the spin-up situation by shedding their extra energy to the lattice. The TLF may introduce the additional inter/intra-effects of spins, which shown in the new maps.

Two aspects should be considered during clarification of the relaxation times; 1) T1 of biological tissues is highly dependent on the Larmor frequency, while T2 is almost independent of this frequency. Therefore, when evaluating the T1 amounts, the magnetic flux density B0 at which the T1 measurement was carried out must be taken into account. 2) relaxation procedures frequently comprise of several parts, so that the explanation with a monoexponential function is merely a crude approximation. By virtue of the significant alterations in the relaxation times of tissue or organs, MR images are likely to be recorded with high tissue contrast, even if the tissue proton densities vary merely slightly from one another. Nonetheless, on the MRI time scale, relaxation procedures of most tissues can be fairly well approximated by a sole exponential function, as shown in Fig. 2. Importantly, adipose tissues like subcutaneous fat structures and bone marrow are an exception, for which minimum two exponential functions are required to be taken into account in order to parameterize the observed relaxation procedures. Our theory may improve this rough approximation by obtaining greater SNR, as shown in Fig. 9. Additionally, the relevant images derived from larger alpha values greater than 2 displayed darker images with more distortion and less detail.

When motion and consequently local field variations in tendons and tissues drop below the Larmor frequency, dipoles straighten in the main magnetic field begin to contribute to T2 relaxation derived from local alterations in the precession ratio. The consequential short T2 relaxation time makes semi-solid tissues like tendons become dark on T2-weighted images. On the other hand, some macromolecules in long T2 fluids, like urine, water, and cerebro-spinal fluid (CSF), seem bright and shiny on T2-weighted images. The TLFmax map may combine the long T1 and T2 along with the short T1 and T2 together. Hence, darkness and signal loss from T2-weighted images in calculus, teeth, and cortical bone, as opposed to ligaments and tendons, are mainly due to low levels of water. The water found in bones, calculus, and teeth typically binds to collagen with a very short T2 relaxation time and subsequently seems dark. Meanwhile, there are slight differences in sensitivity between soft tissue and bone to a dark display at interfaces like between the trabeculae and bone marrow. These interfaces may be shown better in the obtained new maps in our study [42, 43].

The relaxation procedures can be styled via exponential functions with T1 or T2 time constant in a spin system with adequately extreme molecular mobility. The source of the relaxation is varying magnetic fields in an exclusive situation where the protons are located. The attendance of para-magnetic ions may afford a strong relaxation procedure. These para-magnetic ions comprise an un-paired electron and consequently have a magnetic moment by virtue of electron spin which delivers a huge varying field. This shortened T1 time, which depends on a number of factors like the magnetic field power, specific tissue, temperature and the attendance of para-magnetic molecules or ions. These factors may affect the SNR values on the maps. The state of saturation makes the image dark, and the relaxation restores brightness. For attaining T1-weighted images, for instance, cerebral spinal fluid has a long T1 and fat has a short T1, therefore CSF appears dark and fat seems bright. Obtaining an image at a time when the T1 tissues maps are far apart generates an image with great contrast between those tissues. That is called T1 contrast, and the resulting images are named T1-weighted method. Adipose tissue, intrabony lipomas, hemangiomas, degenerative fatty deposits, cysts with proteinaceous liquid, methemoglobin, para-magnetic contrast agents, slow-flowing blood and irradiation variations wholly demonstrate T1-weighted images with strong signal. In contrast, avascular necrosis, infection, infarction, tumors, cysts, sclerosis, calcification and tissues from cortical bone characteristically exhibit a weak signal. Tissues with signal cancellation or suppression contain: calcification, gas or air, tendons, fast flowing blood, scar tissue, and cortical bone [31]. Today, the modern software permits for post-processing of alterations in TR, TE, and TI times, making it likely to capture multiple images, comprising parametric images (T1 FLAIR, T2 FLAIR, STIR, PD, T1, T2) in a considerably shorter period [44,45,46,47,48,49]. Our results are able to characterize these tissues using combination of the values of T1 and T2 at various α-values, as shown in the αmin and αmax maps.

Regularly, T2 does not alter much but T1 increases as the field power rises for most tissues. This issue can be realized through considering the chief technique accountable for the T1 and T2 relaxation time ratios for 1H nuclei, and also an axiom derived from dipole–dipole interactions. When the molecular spinning ratio is nearly equal to the Larmor frequency, the T1 relaxation time is the shortest as it called the correlation time. Those molecules that swirl slower or faster have a longer T1 relaxation time and also have less impact on longitudinal relaxation. Meanwhile, the T2 relaxation mechanism amalgamates all the T1 relaxation procedures. Furthermore, while molecular movement reduces underneath the Larmor frequency, the T2 relaxation takes place in a common style with no energy swapping or T1 exchange. Therefore, bound water molecules, macromolecules, and solids do not revolve rapidly and possess short T2 amounts. Accordingly, the T2 relaxation rises gradually with the molecular spinning ratio. By pure liquids such as CSF a kind of limitation, the T1 and T2 relaxation times are the same and both last several seconds. The general T1 outcome may be considered as the Goldilock phenomenon. For a T1 that is as short as possible, the molecular spin speed ought to be neither “very short” nor “very long”, but “exact right”. In spinning ratio state, most molecules of free water are inefficient at T1 relaxation time due to their extensive range. Therefore, free water possesses long T1 amounts. Hydration layers of water around proteins are further constrained and a meaningfully larger proportion of 1H nuclei revolve close to the Larmor frequency with congruently shorter T1 amounts. The Larmor frequency is straightly related to the B0 field power, subsequently field power changes alter the T1-sweet-spot of Goldilock. However, it has dissimilar influences on diverse substances. An alteration in the field power for protons in extreme mobile molecules does not tremendously change the proportion of protons stirring at that frequency. Thus, neither T2 nor T1 are greatly influenced, at least over the field range utilized for magnetic resonance imaging. However, for protons in molecules with low or medium mobility, the magnetic field shift to a greater magnitude can considerably reduce the portion of those protons that are able to interact at the larger Larmor frequency. Consequently, the T1 relaxation rises with increasing field power. Hence for most tissues, the obtained T1 levels will roughly double while the field power is increased from 0.3 T to 3 T, and rise by proximately 25% between 1.5 T and 3 T.

The T2 relaxation derived from gradually or statically varying fields is not greatly influenced by a Larmor frequency shift. The T2 relaxation procedure in protons with low and medium mobility, which attends the T1 relaxation procedure, is somewhat lengthened via rising in field power, in parallel with the prolonging of T1. From the other point of view, some procedures of T2 relaxation like molecular diffusion and chemical exchange can essentially be more effectual at greater magnetic fields and consequently lead T2 amounts to decrease. It is particularly realized at 7 T in the brain, where high levels of iron are comprised close to the structures such as substantia nigra. By the mean outcomes across all tissue kinds, it suggests that the T2 relaxation does not actually alter over the range of field powers utilized for ordinary clinical imaging-0.3 T to 3.2 T- but decreases at very high levels fields. Measurements at 11.7 T, 9.4 T, and 4.0 T display a predictable incessant reduction in T1 amounts, respectively, but likewise a 20–50% rise in T2 amounts [50,51,52,53,54,55]. Eventually, T1 and T2 relaxation times were lengthened in almost all brain tumor types, but it was problematic to distinguish tumor cases on the basis of relaxation times solely. Meanwhile, the malignancy indices and T1/T2 rate were not cooperative in distinguishing malignant brain tumors [38], but we obtained the new maps with good SNR based on our theory in which the α-parameter plays a key role because of indicating tiny in details on the objects to predict status of objects likely at the later stages.

5 Conclusions

In this study, an appropriate theory was introduced to obtain the new maps with greater SNR through the hybrid combination of the transverse and longitudinal relaxation times towards the better diagnostics in the medical field. The obtained new maps indicated tiny in objects details to predict objects status at the later stages. Our finding may help to grade the different tissues and estimate the duration of healing trends using assessment of the maps extracted from the proposed theory.